A method for implementing calculus calculation by laser field controlled evolution

Abstract

This invention discloses a method for calculus calculations using controlled laser field evolution, belonging to the field of optical computing technology. This invention constructs a seven-layer calculus solution system and a four-step closed-loop computation process, establishing a dual-time mechanism for initialization modeling and normalized computation. The normalized computation time is constant at O(C), avoiding the problem of computational power explosion. This invention establishes a three-layer engineering function adaptation system, adapting to single-variable and finite-multivariate engineering functions that can be regularized and orthogonally expanded with finite-order truncation. It achieves isomorphic function expansion based on orthogonal laser fundamental frequency and multidimensional coupled secondary fundamental frequency wave function sets. Through modal-independent amplitude scaling and standardized phase shifting, it adapts to freely switchable calculus sub-operators, realizing multi-order, multi-dimensional, and multi-type calculus operations. This invention features a unique mechanism for single-dimensional control and multi-dimensional steady-state maintenance, and independent modal evolution. It is equipped with a dual-mode physical isolation architecture to avoid modal crosstalk. It adopts finite-order harmonic symmetric truncation to balance accuracy and efficiency, and abandons the traditional three-segment redundant computation link. It completes the analytical solution by directly controlling the evolution of the entire optical field, which completely solves the pain points of traditional grid iterative algorithms such as computing power explosion, error accumulation, and poor real-time performance. It also makes up for the shortcomings of existing optical field calculus technology, such as weak universality, poor high-dimensional adaptation, and insufficient stability. It has strong engineering practicality and implementation.

Description

Technical Field

[0001] This invention belongs to the field of optical computing, optical field information processing and engineering numerical calculus calculation technology. Specifically, it relates to an optical computing method that relies on the equivalent mathematical operation mechanism of controlled evolution of laser field to achieve high-precision and fast solutions for various types of engineering functions, higher-order differentials, multidimensional partial differentials, high-precision integrals and multidimensional multiple integrals. It can be widely used in high-end engineering numerical computing scenarios such as industrial simulation, dynamic numerical solution, multi-field coupling analysis, high-precision engineering computing power, and real-time optical computing. Background Technology

[0002] Traditional electronic computers commonly employ mesh iterative algorithms such as finite difference and finite element methods to solve higher-order differentials, multidimensional partial differentials, high-precision integrals, and multidimensional multiple integrals. These algorithms require spatial meshing of the solution domain and rely on point-by-point iteration to approximate the numerical results of calculus. The computational load increases exponentially with the solution dimension, computational precision, and number of operations. Under high-dimensional and high-precision simulation conditions, traditional algorithms suffer from inherent defects such as prolonged computation time, large accumulated errors, easy instability, and insufficient real-time performance.

[0003] Currently, domestic and international research focuses on optical simulation computing, including optical fields, metasurfaces, and photonic chips. Theoretical research and prototype verification of optical field calculus operations are underway. These studies utilize structures such as 4f optical systems, spatiotemporally encoded metasurfaces, and microcomb photonic chips to achieve basic differential and integral operations in specific scenarios. This has preliminarily verified the technical feasibility of parallel optical solutions to calculus problems, demonstrating good computational performance and real-time advantages in simple functions, fixed-order, and single-dimensional computation scenarios. However, existing publicly available technical solutions still suffer from core technological bottlenecks and have not yet formed a mature system suitable for large-scale commercial use: First, the computational architecture is rigid and fragmented, with differential and integral operations using discrete hardware and modulation systems, resulting in low hardware reuse and inconsistent computational rules, making integrated solutions impossible. Second, scenario adaptability is extremely poor, only suitable for fixed-order, single-dimensional, and simple functions, lacking a hierarchical function adaptation system and a clear engineering adaptation boundary, making it unable to adapt to multi-dimensional high-order and complex engineering function operations. Third, high-dimensional computational capabilities are lacking, with no mechanism for unidimensional to multi-dimensional homogeneous expansion, making it impossible to achieve single-dimensional control and multi-dimensional steady-state maintenance. First, it is prone to dimensional coupling interference, making it difficult to accurately match the mathematical definition of multidimensional partial calculus. Second, it has insufficient anti-interference capability, lacking a standardized dual-mode physical isolation system, making it prone to mode mixing and optical field crosstalk in high-order multimodal superposition scenarios, resulting in poor computational stability. Third, it has redundant computation links, generally adopting a three-stage architecture of "forward transformation-computation-inverse transformation," which is redundant in modeling, lacks self-consistency, and lacks a dual-time-sequence operation mechanism. The normalized computation delay increases sharply with parameter upgrades, failing to solve the inherent pain points of traditional algorithms, and there is currently no universal laser field calculus solution that is suitable for all working conditions, high precision, and high stability.

[0004] While existing optical simulation computing technologies have overcome some of the real-time limitations of traditional electronic iterative algorithms, their overall technical system is fragmented and lacks core mechanisms, failing to meet the industrial demands for high-precision, highly adaptable, and highly stable calculus solutions. Current technologies lack a hierarchical and differentiated engineering function adaptation system, resulting in ambiguous adaptation boundaries, incomplete scenario coverage, and an inability to regulate and adapt to various conventional and special engineering functions, making it difficult to adapt to general orthogonal series modeling formats. Simultaneously, the calculus computation system is fragmented and lacks unified rules, lacking a standardized closed-loop architecture with modular decoupling. High- and low-order, single- and multi-dimensional computational logics cannot communicate, leading to extremely poor engineering scalability. Furthermore, existing solutions lack a dedicated mechanism for controlled evolution of the light field, lack independent operator modulation at a single frequency point, and lack hierarchical evolution capabilities, making it impossible to freely switch calculus operators and comprehensively adapt to multi-scenario computational needs.

[0005] In multidimensional computing scenarios, existing technical architectures have weak homogeneous expansion capabilities, and one-dimensional and multidimensional computing systems cannot interoperate. They lack the core mechanism of independent single-dimensional operator control and steady-state maintenance of the optical field in other dimensions, and cannot accurately fit the mathematical definitions of multivariable partial calculus and multiple iterated integrals. High-dimensional computing suffers from severe coupling interference and low solution accuracy. At the same time, there is currently no standardized and comprehensive modal anti-crosstalk system in the industry, and there is no distinction between targeted isolation and global isolation conditions. Under the conditions of high-order harmonics and multidimensional coupled mode superposition, modal mixing and optical field crosstalk problems are prominent, and the error accumulation effect is significant. The adaptability to high-precision and high-stability engineering solution scenarios is seriously insufficient.

[0006] Furthermore, existing optical calculus techniques lack a standardized four-step closed-loop optical field evolution rule, missing a complete set of underlying mechanisms for single-mode differentiated modulation, hierarchical controlled evolution, and global analytical direct reading, thus failing to achieve multi-parameter and multi-scenario calculus adaptation. Simultaneously, they lack a dual-temporal architecture that distinguishes between initialization modeling and normalized computation; the latency of normalized computation increases significantly with increasing dimension, order, and number of operations, failing to fundamentally address the industry pain points of traditional grid iterative algorithms, such as computational power explosion, poor real-time performance, and error accumulation. In summary, the field urgently needs a standardized closed-loop laser field calculus solution system with clearly defined engineering adaptability boundaries, modular decoupling, multi-dimensional homogeneous expansion, dual-mode anti-crosstalk, and efficient dual-temporal operation. This system would break through existing technological bottlenecks and promote the large-scale application of optical simulation computing in high-end engineering simulation and high-precision numerical solution. Summary of the Invention

[0007] 1. Definition of core terms To unify the technical terminology throughout the text, eliminate semantic ambiguity, meet the requirements of sufficiency of patent disclosure, and solidify the technical boundaries and scope of protection of this invention, this invention provides unified legal definitions for core proprietary technical terms that are repeatedly used throughout the text. All technical expressions in the text shall conform to these definitions, as follows: (1) Meta, Dimension, Order, Suborder, and Weight (Basic Core Definitions): To unify the technical terminology throughout the text and eliminate semantic ambiguity, this invention exclusively defines five independent underlying concepts. All technical descriptions throughout the text strictly adhere to these definitions. These five concepts are independent of each other, have no overlap, and each performs its own function, corresponding to variable unit, spatial scale, modeling accuracy, calculus level, and integral level, respectively. Meta: refers to the independent variable in the modeling and solving process. It is the smallest independent variable unit for function modeling and calculus operations in this invention. The number of variables directly determines the complexity of the function to be solved. Dimension: refers to the physical expansion spatial dimension corresponding to the independent variable. It is uniformly quantified using the letter D, where D is a finite positive integer. The multidimensional, multivariate coupled computational space is used to precisely define the spatial scale of laser light field modeling, adapting to multivariate function modeling scenarios. Order: refers to the harmonic mode truncation order of the orthogonal series expansion, uniformly quantified by the letter M. M is a finite positive integer that can be configured, uniquely determining the fineness of the light field modeling and the accuracy of calculus engineering solutions. The larger the value of M, the more comprehensive the harmonic coverage and the higher the computational accuracy. Degree: a differential-specific hierarchical parameter, defined precisely and unambiguously throughout the text using a dual-caliber approach: for univariate function differential scenarios, n is uniformly used to represent the universal differential degree; for multidimensional, multivariate function partial differential scenarios, n is uniformly used... The degree of the independent differential of the d-th dimension variable is used. The differential levels of each dimension are independent and can be configured differently. The higher the degree of the differential, the higher the level of the differential calculation in the corresponding dimension. During the operation, the amplitude of the light field is proportionally amplified by the nth power of the corresponding fundamental frequency point, and the standardized phase shift of the corresponding degree is matched synchronously. "Number of integrals" is a parameter specific to the integral. The entire paper adopts a dual-caliber precise adaptation and unambiguous definition: for univariate functions, n is used to represent the unified integral number; for multivariate functions, n is used to represent the integral multiplicity. The integral multiplicity of the d-th dimension variable represents the independent integral multiplicity. The integral multiplicity of each dimension is independent and can be configured differently. The higher the integral multiplicity, the higher the cumulative level of the corresponding dimension. During calculation, the corresponding fundamental frequency point is used. The reciprocal of the power is the scaling factor to achieve inverse proportional attenuation of the light field amplitude, and synchronously match the standardized phase shift of the corresponding number.

[0008] (2) Fundamental frequency (fundamental frequency mode): In the orthogonal series system, the discrete orthogonal reference frequency light wave used to construct the laser light field, complete the isomorphic expression of functions and the equivalent operation of calculus is the underlying core component for single-variable univariate function and multivariate multidimensional function light field modeling. In the one-dimensional modeling scenario, the fundamental frequency is a single-dimensional independent orthogonal reference frequency, which can be directly adapted to single-variable function modeling and calculus operation; in the multidimensional multivariate modeling scenario, each dimension independently configures its own orthogonal exclusive fundamental frequency according to its own periodic characteristics. After coupling, the single-dimensional fundamental frequencies of each dimension completely generate multidimensional sub-fundamental frequency composite modes. There are no independent single-dimensional fundamental frequencies remaining in the multidimensional operation domain. Only the coupled sub-fundamental frequency composite modes participate in the operation, which is adapted to the needs of multivariate function coupling modeling and multidimensional calculus solution. The fundamental frequency distribution law of this invention achieves bidirectional adaptive matching based on the periodic characteristics of the corresponding dimension of the function to be solved and the orthogonal series expansion method. The fundamental frequency light waves of each order are orthogonal to each other and do not crosstalk each other. Each frequency point corresponds to an independent optical path and an independent mode, supporting independent modulation and independent optical field evolution.

[0009] (3) Sub-fundamental frequency (multi-coupling sub-fundamental frequency mode): In the multi-dimensional multivariate function modeling scenario, the composite orthogonal mode generated by the mutual coupling and superposition of multi-dimensional fundamental frequencies is fully matched with the construction of the multivariate function series coupling basis, and is used for multi-dimensional operation scenarios.

[0010] (4) Frequency point: refers to the position of the independent orthogonal optical field mode corresponding to a single fundamental frequency mode, and is uniformly represented by mathematical symbols. Quantitative characterization shows that the frequency points of this invention correspond one-to-one with the fundamental frequency mode order, representing the smallest physical unit for single-mode independent operator modulation. In this invention, under a univariate scenario... Multi-dimensional scenarios The frequency distribution gradient and distribution rules can be matched with the orthogonal expansion of the function to be solved in a two-way adaptive manner, so as to achieve accurate modeling of different orthogonal series systems and different function forms.

[0011] (5) Mode: refers to the steady-state distribution of independent optical paths corresponding to a single fundamental frequency and a secondary fundamental frequency. It has independence and modulatorability. Among them, the single-dimensional independent fundamental frequency modes are orthogonal to each other, while the secondary fundamental frequency composite modes generated by multi-dimensional coupling do not necessarily have orthogonality. In this invention, all optical field evolution, operator action, and physical isolation are based on the mode as the smallest execution unit.

[0012] (6) Wave function: refers to the basis function describing the spatial and phase distribution of the laser light field. It is the core carrier for the optical field physical modeling and functional isomorphic expression in this invention. All engineering functions are represented in the entire spatial domain through the linear superposition of wave functions. The mathematical form of the standardized one-dimensional laser basis wave function in this invention is: The corresponding D-dimensional single-basis wavefunction expansion is: Representing the d-th dimension space, the d-th dimension space There are independent orthogonal fundamental frequencies; the total number of dimensions is D, and the single-dimensional harmonic truncation order is M. The wave functions corresponding to the independent single-dimensional fundamental frequencies in each dimension are pairwise orthogonal. The sub-fundamental frequency composite wave functions generated by multi-dimensional coupling do not necessarily have orthogonality, forming a complete set of standardized orthogonal bases.

[0013] (7) Wave function set: This is the standard expansion basis set for the light field modeling of this invention. It is the core mathematical carrier for engineering function expansion and controlled evolution of the light field. It is divided into two types according to the solution dimension, which are adapted to single-variable and multi-variable modeling scenarios respectively: one type is a single-dimensional single-variable orthogonal fundamental frequency wave function set, which is adapted to the solution scenario of univariate function. The frequency points are arranged in an orderly manner, each mode is orthogonal to each other, and there is no crosstalk. It can be standardized by using a finite-order symmetric truncation system as needed; the other type is a multi-dimensional multivariable coupled secondary fundamental frequency wave function set, which is generated by coupling single-dimensional orthogonal fundamental frequency wave functions of each dimension. It accurately matches the construction of multivariate function series basis and is exclusively adapted to the solution scenario of multivariate calculus. Among them, the single-dimensional single-variable orthogonal fundamental frequency wave function group has complete orthogonal characteristics, while the multi-dimensional coupled secondary fundamental frequency wave function group is a composite mode group generated by orthogonal fundamental frequency coupling in various dimensions. It does not necessarily have orthogonal characteristics, but only precisely matches the construction of the multivariate function series coupled basis, which is suitable for multi-dimensional multivariate calculus solution scenarios. It can achieve standardized modeling of engineering functions and isomorphic expression of light fields through finite-order truncation expansion.

[0014] (8) Amplitude and phase reference calibration (amplitude normalization, phase zeroing): This is a pre-calibration method for physical mapping exclusive to this invention. It belongs to the standardized reference calibration mechanism for single-dimensional orthogonal fundamental frequencies in the laser physical field initialization stage. The secondary fundamental frequencies generated by multi-dimensional coupling do not need to be calibrated separately. They are generated by coupling of the orthogonal fundamental frequencies of each dimension that have completed the reference calibration. By normalizing the amplitude, the amplitude of all orthogonal fundamental frequency optical fields is uniformly calibrated to unit 1, the initial amplitude difference of each fundamental frequency beam is regulated, and the initial amplitude reference of all fundamental frequency mode optical fields is unified in the whole domain. By zeroing the phase, the phase zeroing reference of the whole domain fundamental frequency optical field is unified, and the initial phase of all orthogonal fundamental frequency optical fields is uniformly anchored at the extreme position of the light wave peak or trough, completing the whole domain fundamental frequency phase reference regulation, establishing a unique and traceable initial phase extreme value reference, and providing a unified physical reference for subsequent one-dimensional scene phase polarity discrimination and standardized phase control. This pre-calibration mechanism only performs standardization and regularization on the underlying orthogonal fundamental frequency modes, which can eliminate the initial amplitude deviation and phase offset error of the optical path hardware, regularize the initial state of the fundamental frequency modes in the entire domain, and the secondary fundamental frequency modes subsequently coupled in naturally inherit the calibrated amplitude and phase reference, providing a unified, standardized, and traceable standardized physical reference for subsequent series coefficient encoding, amplitude scaling modulation, standardized phase offset control, and amplitude and phase joint mapping.

[0015] (9) Orthogonal decoupling and zeroing: The core orthogonal underlying mechanism of this system is that each order frequency point and mode corresponds one-to-one and is orthogonal to each other, and is not affected by the dimension. It can realize independent operation and independent zeroing of single-order modes, and completely avoid modal crosstalk and coupling interference. In multi-dimensional scenarios, the independent fundamental frequencies of each dimension have no spontaneous coupling characteristics. It is necessary to accurately match the multivariate function series basis construction through the system multi-dimensional coupling modeling and parameter modulation to generate standardized secondary fundamental frequency modes. The overall multi-dimensional coupled modes cannot be decoupled in the whole domain, but the single-dimensional operation rules and operator control logic are completely independent and controllable. It can perform calculus operator evolution on the target dimension alone, and the other dimension modes remain in steady state. It strictly fits the mathematical definition of multivariate multiple partial differential and multiple accumulated integral, and ensures the independence and solution accuracy of multi-order and multi-dimensional partial differential operations.

[0016] (10) Encoding: The optical field mapping mechanism for establishing the correspondence between series coefficients and physical parameters of the laser optical field in this invention does not restrict the unique encoding form. According to the operation scenario, it is divided into two types of adaptable mechanisms: amplitude encoding and amplitude-phase joint encoding. The two are mathematically similar but adaptable to different scenarios. The first type: amplitude encoding, is a general adaptation method, applicable to multidimensional operation and conventional one-dimensional operation scenarios. It only maps the absolute value of the series coefficients to the optical field amplitude, and the positive and negative signs of the coefficients are identified by the peripheral algebra. The second type: amplitude-phase joint encoding, is a feasible implementation method adapted to the one-dimensional operation scenario. It adopts the amplitude and phase joint modulation method. The absolute value of the series coefficients is loaded by optical path amplitude modulation, and the optical path phase loading of the positive and negative polarities of the coefficients is realized by reference phase and π offset phase modulation, which completely covers all information of the series coefficients. After the optical field evolution is completed, the optical field phase state is directly read. The operation polarity is determined by the differential modulation results of the reference phase and π offset phase. The calculus operation result with positive and negative signs is directly output. There is no need for post-coefficient multiplication and compensation operation. The operation link is simple and efficient.

[0017] (11) Operator: The core operation regulation unit of the global standardized calculus of this invention has completely unified rules for one-dimensional and multi-dimensional differential operators. The integral operator is uniformly divided into two general regulations: zero frequency without target dimension and zero frequency with target dimension. It does not distinguish between one-dimensional and multi-dimensional scenarios. The operation rules of the whole domain are consistent and unified. The specific regulations are as follows: Differential operator: For the target dimension frequency point mode, the amplitude is proportionally amplified by the nth power of the frequency point corresponding to the differential number n, and matched with n times the positive direction. Standardized phase shift enables independent modal differential evolution; Integral operator: For all fundamental and secondary fundamental modes excluding the target dimension 0 frequency, the amplitude attenuation factor is inversely proportional to the inverse power of the frequency point corresponding to the integral multiplicity n, and matched with n times the inverse phase shift. Standardized phase shift enables modal-independent integral evolution; dedicated operator for zero-frequency integration in the target dimension: for all fundamental and secondary fundamental modes containing zero-frequency in the target dimension, according to... The optical path amplitude is adjusted by a factor while the phase remains unchanged, thus uniformly adapting to integration calculation scenarios in all dimensions, including zero-frequency components of the target dimension.

[0018] (12) Modal Independent Modulation: This refers to the fact that each fundamental frequency and coupled secondary fundamental frequency mode in each dimension is matched with a dedicated calculus operator to achieve one-to-one independent and controlled evolution. Differentiation and integration have differentiated and fixed optical path amplitude and phase modulation rules. Each mode is independently modulated and does not crosstalk with each other, which can completely avoid mode coupling interference. The present invention has a unified operation rule across the entire domain. Univariate and multidimensional partial differentials and multiple integrations are all completed independently by standardized operators. Differentiation relies on the frequency point power to amplify the amplitude and match the positive phase shift. Integration relies on the frequency point power to attenuate the amplitude and match the negative phase shift. Zero-frequency modes do not participate in optical field evolution. The corresponding compensation terms are completed by electronic closed-loop operation, realizing the modal independent and controlled evolution operation of full-order and full-dimensional calculus.

[0019] (13) Single-dimensional control and multi-dimensional steady-state maintenance: The core mechanism of multi-dimensional multivariate partial differential and multiple accumulated integral is that after the multi-dimensional light field is coupled by the one-dimensional wave function of each dimension, it is standardized to generate multi-dimensional coupled secondary fundamental frequency components. The initial single-dimensional fundamental frequency completely exits the operation link and there is no independent single-dimensional fundamental frequency residue. The multi-dimensional operation relies on the independent secondary fundamental frequency composite mode to complete the controlled evolution with the exclusive independent calculus operator of each dimension. Under the multi-mode superposition condition, there is a natural risk of crosstalk and aliasing. It is necessary to match the dual-mode physical isolation architecture as needed to avoid operation distortion. During the operation, the calculus operator is loaded on the coupled basis component corresponding to the target solution dimension to achieve controlled evolution. The coupling mode parameters corresponding to the other dimensions remain unchanged throughout the process. It strictly fits the mathematical definition of multivariate multiple partial differential and multiple accumulated integral with respect to a single variable and the other variables remain unchanged. It is the core guarantee mechanism for the accurate solution of multi-dimensional multivariate multiple partial differential and multiple accumulated integral.

[0020] (14) Finite-order truncated expansion: Within the allowable range of engineering accuracy, a finite positive and negative symmetric order is used to replace the infinite series expansion. The truncation order M is used to balance the solution accuracy and computational efficiency, which is the core modeling paradigm for the engineering implementation of this invention. For multidimensional multivariate function solution scenarios, each dimension can be independently configured with a different truncation order according to its own solution accuracy requirements to adapt to non-uniform accuracy solution conditions. In order to unify the description throughout the text and simplify the modeling and calculation logic, the entire text of this invention uses the same truncation order M to complete the global series truncation expansion in multidimensional multivariate operation scenarios. This equivalent processing method is completely consistent with the mathematical essence of the differentiated order of each dimension and does not change the core operation mechanism.

[0021] (15) Standardized phase shift: The regularized phase control mechanism corresponding to the calculus operator of this invention, the differential operation corresponds to Positive multiple phase shift, corresponding to integral operation The phase shift is a reverse multiple, and the shift rule is uniform across the entire domain and can be extended by repetition or restatement.

[0022] (16) Dual-time-sequence operation mechanism: A two-layer general-purpose operation architecture that distinguishes between the initialization modeling time sequence and the normalized operation time sequence, while taking the multidimensional partial differential equation solution scenario as the core and adapting to the one-dimensional basic operation condition. Among them, the initialization modeling is a one-time overhead shared by univariate and multidimensional functions, including pre-processes such as function regularization, series coefficient solving, and light field isomorphic representation; the normalized evolution and reading time sequence have dimensional differences, and the overhead of the one-dimensional and multidimensional partial differential equation normalized time sequence is: There is no problem of computing power explosion. Only the multidimensional integral electronic compensation stage has a controllable and limited computing power overhead, and the time series is stable and controllable throughout the domain.

[0023] (17) Modal aliasing / optical field crosstalk: The amplitude distortion and phase coupling interference problems generated under multi-mode superposition conditions exhibit different characteristics in different dimensions. In one-dimensional computing scenarios, there is only the risk of crosstalk aliasing between fundamental modes of the same order and between high and low order, without cross-dimensional coupling interference; in multi-dimensional computing scenarios, in addition to single-dimensional modal crosstalk, the sub-fundamental frequency composite modes generated by multi-dimensional fundamental frequency coupling are more prone to cross-modal and cross-dimensional coupling aliasing distortion, which is the core problem of the entire domain that the dual-mode physical isolation architecture of this invention specifically addresses.

[0024] (18) Pure Algebraic Scale Correction: For coordinate ratio deviations caused by scale substitution and periodic regularization of one-dimensional and multi-dimensional functions, a simple post-correction method is used to restore the true value of the operation by fixing the algebraic power ratio. There is no cross-domain transformation or decoding operation. It is suitable for single-variable coordinate scale substitution correction in one-dimensional scenarios; it is suitable for multi-dimensional synchronous scale normalization and periodic substitution in multi-dimensional scenarios. It can synchronously complete algebraic correction for independent scale ratio deviations in each dimension, accurately adapt to the true value restoration requirements of functions after one-dimensional and multi-dimensional regularization, and is a lightweight correction mechanism that is universally applicable.

[0025] (19) Dual-mode physical isolation architecture: The invention has a dedicated anti-crosstalk protection system, which includes two types of standardized isolation solutions that can be implemented independently: targeted physical isolation and full-domain full-coverage physical isolation. With multi-dimensional computing scenarios as the core, it is adapted to uni-level basic computing scenarios and can effectively avoid the problems of mode mixing and light field crosstalk caused by multi-modal superposition, ensuring the accuracy and stability of light field evolution and computing.

[0026] (20) Computing power operation architecture: The present invention adopts a general-purpose and adaptable optical main operation combined with electronic auxiliary compensation collaborative computing power operation mode, which is differentiated to one-dimensional and multi-dimensional partial calculus operation scenarios and computing power logic. One-dimensional operation is mainly based on pure optical field parallel operation, and the computing power overhead of electronic compensation is extremely low; multi-dimensional operation relies on the independent evolution of single-dimensional optical field and the multi-dimensional steady-state maintenance mechanism, and the oscillation mode of the calculus optical field is constant throughout the process. The timing operation and boundary nesting compensation terms are completed by electronic system to assist in the completion, which fits the physical operation boundary of one-dimensional and multi-dimensional differentiated light field, and ensures the stability of calculus solution in all dimensions and under all working conditions. This computing power adaptation method is a practical auxiliary means and is not the core innovation of this invention.

[0027] (21) Off-domain mapping fallback adaptation: For special engineering functions that cannot be conventionally regularized in one-dimensional or multi-dimensional domains, a full-domain fallback mechanism is established by relying on cross-domain and inverse mapping techniques known in the field to complete the pre-adaptation modeling and result restoration. One-dimensional special single-variable functions can be normalized and adapted to orthogonal series through off-domain mapping; multi-dimensional complex multivariate coupled functions can achieve high-dimensional function domain space transfer through cross-domain mapping to adapt to the modeling requirements of multi-dimensional composite modal series expansion. This method is a well-known technology in the industry and belongs to the auxiliary means of engineering function regularization fallback. It does not participate in the one-dimensional and multi-dimensional core optical field operation links and is not the core innovation of this invention.

[0028] (22) Polarity (coefficient polarity / operation result polarity): refers to the positive or negative attribute of the orthogonal series expansion coefficients and the calculus operation result. It is the core identifier parameter for one-dimensional and multi-dimensional optical field amplitude encoding and operation result interpretation. The universal polarity determination mechanism of this invention: amplitude encoding only maps the absolute value of coefficients. The positive or negative value of coefficients is an independent algebraic identifier on the periphery and does not participate in the physical encoding modulation of the optical path. The single optical path first relies on its own optical field phase evolution and offset to complete the basic polarity determination internally, and then combines the peripheral positive and negative algebraic identifiers to output the final signed result. Among them, the unidimensional single-dimensional fundamental frequency optical field has a unique steady-state spatial distribution shape, and the extreme value feature corresponding to the phase offset is unique. It can directly complete the unambiguous and autonomous polarity determination based on the extreme value feature of the static waveform of the optical field. The multi-dimensional coupled sub-fundamental frequency optical field is a multi-dimensional phase superposition composite mode. The true value of the physical field is uniquely determined, but there is no unique static extreme value criterion for directly determining the positive or negative value of coefficients, and there is naturally ambiguity in polarity discrimination. The one-dimensional core link of this invention can autonomously complete accurate polarity determination. Multi-dimensional polarity correction relies on well-known optical discrimination methods in the field as a backup. Well-known auxiliary technologies are only used for multi-dimensional polarity result correction and do not interfere with the one-dimensional and multi-dimensional general core technology links of this invention. They are not within the scope of the core innovation of this invention.

[0029] (23) Series Expansion: The standardized modeling paradigm of this invention (the series expansion and conventional series expansion mentioned throughout this paper all refer to orthogonal series expansion, and are semantically equivalent) is a general orthogonal series modeling and optical field isomorphic representation carrier adapted to one-dimensional single-variable functions and any finite D-dimensional multivariate functions, realizing a complete unification of one-dimensional and multi-dimensional orthogonal modeling paradigms and technical approaches. For one-dimensional single-variable functions, the wave function corresponding to the single-dimensional independent orthogonal fundamental frequency is used as the basis to complete the finite-order orthogonal series expansion and function equivalent representation. The single-dimensional fundamental frequency basis has complete orthogonal characteristics. For multi-dimensional multivariate functions, the periodic characteristics of each independent variable are matched with a dedicated orthogonal fundamental frequency. The single-dimensional orthogonal fundamental frequencies corresponding to each variable are independent of each other and each satisfies the orthogonal characteristics. The orthogonal fundamental frequencies of all variables are coupled to form a multi-dimensional composite orthogonal basis set. Based on this coupled basis, the orthogonal series expansion modeling of multivariate functions is completed, accurately matching the construction law of the coupled basis of multivariate functions. It should be clarified that orthogonal series expansion is a mature and well-known mathematical tool in the field and does not fall within the scope of the core innovation of this invention. The core innovation of this invention is the original light field modeling architecture adapted to this unified expansion paradigm, the controlled evolution mechanism, and the one-dimensional and multi-dimensional unified calculus solution closed-loop system, which breaks the barrier of the separation between the traditional one-dimensional and multi-dimensional modeling paradigms and realizes the unification of the one-dimensional and multi-dimensional orthogonal series expansion format and the self-consistency of the mechanism.

[0030] 2 Overall Inventive Concept 2.1 Overall Inventive Concept This invention relies on the applicant's prior application for a four-step framework for isomorphic computation of general physical fields. Based on this general framework, it uniquely establishes a standardized four-step isomorphic evolution method system and closed-loop technical architecture adapted to specific calculus solution scenarios. This involves sequentially constructing a laser field state evolution space adapted for calculus computation, isomorphically representing the laser physical field of the calculus function to be solved, performing controlled evolution of the laser physical field to correspond to equivalent calculus numerical computation, and reading the calculus computation results from the target state of the laser physical field. Combining the light field computation logic of the applicant's prior application for vector numerical light computation patent, the core innovation of this invention is the construction of a standardized, seven-layer, all-dimensional, and comprehensive calculus solution system. This system achieves standardized calculus solution capabilities across multiple scenarios, parameters, orders, and dimensions through seven levels: function adaptation, order solving, variable dimensions, computational architecture, operator switching, multiple unifications, and full-domain analytical direct reading. This invention overcomes the shortcomings of traditional technologies, such as rigid processes, weak barriers, and easy circumvention. It constructs a complete closed-loop technology barrier with a seven-layer closed-loop technology architecture: "Top-level dedicated solution architecture - layered physical kernel mechanism - multi-dimensional modular decoupling - dual-mode anti-crosstalk protection - dual-time-series high-efficiency operation - full-domain accuracy guarantee - multi-condition standardized output". It solidifies exclusive underlying physical operation rules such as single-frequency point independent dedicated operator modulation, single-dimensional directional control and multi-dimensional steady-state maintenance, multi-element coupling sub-fundamental frequency mode adaptation, dual-mode standardized physical isolation, and three-layer differentiated layered function adaptation. It builds a hard-core technology barrier deeply solidified at the physical mechanism level, and establishes a multi-faceted, multi-level, and systematic dedicated technology protection system. It significantly improves the accuracy and dedicated adaptability of technology protection, and makes up for the industry shortcomings of traditional optical computing technology protection, such as coarse granularity, weak technical barrier layers, and vague dedicated protection boundaries.

[0031] This invention overcomes the limitations of traditional optical computing technology's single-function adaptation. Based on the inherent physical property that engineering computing systems can process finite-domain information, it establishes a three-layer differentiated, self-consistent closed-loop engineering function adaptation system: precise matching, conventional preprocessing regularization, and out-of-domain mapping as a fallback. It clearly defines the unique function adaptation boundary of this invention, adapting to engineering-solvable functions within a defined range, without restricting the original function's function form, periodicity, convergence characteristics, or expansion method. The specific layered adaptation mechanism is as follows: The first category is the exact matching adaptation function. This type of function naturally matches the system's laser orthogonal wave function in terms of its own wave characteristics and spatial wave features. It natively satisfies the strict expansion conditions of finite-order discrete orthogonal series, requiring no preprocessing, scaling, or auxiliary transformation tools. It can be directly and accurately represented based on the finite-order orthogonal series system. Within the engineering accuracy range corresponding to finite-order harmonic truncation, it can directly achieve error-free and accurate calculus solutions. After computation, the results can be directly read and output without any correction or inverse transformation decoding operations.

[0032] The second category comprises functions that can be adapted through conventional preprocessing. This category includes two main types of commonly used engineering functions: intrinsically periodic functions and aperiodic functions in finite / infinite fields. These functions do not meet or fully fit the standard orthogonal series modeling conventions of the system, but can be adapted and adapted using industry-standard and well-known mathematical preprocessing methods. For periodic functions, standardization and adaptation can be achieved through period matching, scale substitution, and periodic normalization. For aperiodic functions, equivalent normalization can be achieved through effective feature interval truncation, artificial equivalent period construction, and scale normalization. Both types of normalization methods are industry-standard, minimally simplistic preprocessing techniques. Their purpose is to perform morphological adaptation on the original functions to ensure they meet the mathematical conditions for orthogonal series expansion of the system, thereby achieving finite-order orthogonal series standardization expansion and isomorphic representation of the light field. After preprocessing and regularization, and orthogonal series expansion modeling, the modal morphology of this type of function is completely matched with the orthogonal fundamental frequency system of the system. After the subsequent optical field evolution calculation is completed, the true value of the calculation can be restored by pure algebraic scale multiplication correction. There is no need to rely on inverse transform decoding. The accuracy is controllable and the solution is stable throughout the process.

[0033] The third category is the out-of-domain mapping fallback adaptation function. This type of function has complex characteristics and cannot be modeled and adapted for orthogonal series expansion using conventional preprocessing methods such as interval truncation, periodicity reduction, and scale substitution. It belongs to a niche, special engineering function category. This invention configures an out-of-domain mapping fallback adaptation mechanism for this type of function. It can rely on public dedicated domain mapping tools to complete pre-adaptation modeling, perform cross-domain mapping and morphological reduction processing on complex primitive functions, enabling them to complete domain space migration and meet the mathematical conditions for orthogonal series expansion in this system. This transforms them into the standard form of orthogonal series adapted to this system, completes standardized local orthogonal series expansion and isomorphic expression of the light field, and achieves stable calculus solutions. Correspondingly, this type of function relies on out-of-domain cross-domain mapping for pre-modeling and expansion, involving a real domain migration process. After computation, the true value cannot be restored through pure algebraic correction; it must rely on a matching inverse mapping decoding method to restore the final result output. This serves as a fallback guarantee mechanism for the full-scene solution capability of this invention, completing the solution adaptation and expansion modeling closed loop for special functions.

[0034] The aforementioned three-layer differentiated hierarchical adaptation mechanism clarifies the unique legal adaptation boundary of this invention: within the allowable range of engineering accuracy, all single-variable and finite-multivariable engineering function calculus operation scenarios that can be normalized through conventional mathematical methods such as direct expansion, substitution, interval truncation, and periodic normalization, or normalized through dedicated mapping, and achieve finite-order truncated expansion of orthogonal series based on wavefunction groups, fall within the adaptation scope of this invention; operation scenarios that cannot achieve finite-order orthogonal series expansion or cannot achieve engineering truncation convergence are not within the adaptation scope of this invention. This invention uniformly adapts multiple types of engineering functions within a limited range into finite-order orthogonal series expansion models, relying on controlled evolution laser fields to achieve high-precision and stable calculus solutions. It possesses seven complete core capabilities, comprehensively distinguishing itself from traditional numerical calculus solution techniques, as detailed below: First, the function adaptation scenarios are broad. Relying on a three-layer differentiated function adaptation strategy of precise matching adaptation, conventional preprocessing regularization adaptation, and out-of-domain mapping fallback adaptation, it can adapt to multiple types of periodic, aperiodic, decaying, and time-varying engineering functions, and is compatible with multiple types of orthogonal series expansion forms with orthogonal completeness. For engineering functions within the adaptation range, the preferred modeling method is to use finite-order orthogonal series with symmetric full-order truncation in the range of [-M to M].

[0035] Second, the order solution is flexible. Relying on the decomposable modulation characteristics of the spatial modal calculus operator, it can accurately adapt to the solution of each derivative and multiple integral within the accuracy constraint range of the finite harmonic order M of this invention without iterative fitting. The accuracy level can be set as needed by the configurable finite positive integer order M. The larger M is, the higher the solution accuracy. It can adapt to multiple engineering conditions such as lightweight fast solution, conventional accuracy solution, and ultra-precise high-order solution.

[0036] Third, it boasts strong scalability across variable dimensions. This invention possesses lossless adaptability and scalability from one dimension to higher dimensions, with all underlying mathematical formulas and operational rules maintaining a consistent and unified paradigm. Relying on independent orthogonal optical paths and independent fundamental frequency configurations in each dimension, it achieves independent controllability of single-dimensional operators and steady-state maintenance in multiple dimensions. The multidimensional optical field adaptively generates coupled secondary fundamental frequency components based on the coupling characteristics of multivariate functions, fully matching the construction of multivariate series basis. It can accurately apply multivariate multiple partial differential and multiple iterated integral operators to the target dimension alone, while maintaining the steady-state of the optical field in other dimensions. It strictly conforms to the mathematical definitions of multivariate multiple partial differential and multiple iterated integral in multiple dimensions, supporting expansion from one dimension to any finite D-dimensional scene. The normalized operation timing is... .

[0037] Fourth, the computational system is unified. Differential and integral operations rely on a unified mathematical modeling and evolution system for laser light fields, sharing the same set of physical field computational paradigms and underlying operating rules. This achieves an integrated and homogeneous design of the theoretical architecture for the two types of operations, completely solving the problems of fragmented and inconsistent models in traditional calculus computational systems.

[0038] Fifth, the operational operators can be switched freely. Based on the unified mathematical operation paradigm and the theoretical framework of light field evolution, the configuration of differential and integral operators can be flexibly switched as needed, adapting to the needs of differential and integral operations in both directions, realizing full-scenario compatibility and adaptation of calculus operators, and greatly expanding the scope of engineering applications.

[0039] Sixth, the multiple operation paradigms are completely unified. This invention employs a completely consistent optical field evolution operation architecture and standardized modulation process for arbitrary finite-degree multiple differentiation and arbitrary finite-multiple-fold multiple integration. There are no fundamental differences between high-order and low-order operations, or between multiple integrations and single integrations, at the operational paradigm, operating mechanism, and physical link levels; only the operator order parameters are differentiated. This completely distinguishes itself from the inherent defects of traditional numerical algorithms, such as the fragmented high- and low-order operation processes, the need to reconstruct algorithm logic for repeated improvements, and the requirement for iterative architecture modifications. It achieves calculus order expansion without process changes, system reconstruction, or adaptation costs, resulting in a highly unified multidimensional operation paradigm and extremely strong scalability.

[0040] Seventh, it enables a single-step global analytical evolution of the light field and precise direct reading at any point. Unlike the traditional numerical algorithm's solution paradigm of grid-by-grid iteration and point-by-point discrete fitting, this invention generates a global continuous steady-state analytical light field through a single complete controlled evolution of the standardized light field, forming a complete and continuous analytical solution for calculus operations, rather than a discrete approximate numerical solution. After the light field evolution is complete, the precise calculation results at any coordinate point in the spatial domain can be directly read without discrimination, iteration, or quadratic fitting. This completely eliminates the grid truncation error, iteration accumulation error, and point interpolation distortion problems of traditional discrete solutions, achieving high-precision analytical-level true value output. This is one of the core disruptive features that distinguishes this invention from existing numerical calculus solution techniques.

[0041] Based on this, this invention establishes a global dedicated optical path read / write mechanism adapted to one-dimensional orthogonal fundamental frequency and multi-dimensional coupled secondary fundamental frequency modes. It configures two independent, separately deployable standardized physical isolation protection architectures, both falling within the scope of this invention: the first is targeted physical isolation, precisely isolating fundamental frequency components and multi-dimensional coupled secondary fundamental frequency components with crosstalk risks; the second is global, full-coverage physical isolation, uniformly isolating all fundamental and secondary fundamental frequency modes throughout the entire process, comprehensively avoiding modal crosstalk risks. This invention reuses the applicant's prior patent application for vector numerical optical computation, using parallel computation logic of the optical field matrix to solve for orthogonal series coefficients, completing amplitude scaling and phase shift physical modulation in the spatial optical field, equivalently achieving integrated differential and integral operations. Simultaneously, it employs a dual-time mechanism separating initialization modeling and normalized iteration, ensuring a constant time overhead for normalized optical field computation. To address the issue of missing terms in the integral operation due to the inability to solve for boundary compensation in the optical field, a precise closed-loop pure numerical electronic calculation method without iteration or fitting is adopted to fill in the missing terms. This method differs from the traditional electronic integration method of simulation fitting and grid iterative approximation, eliminating fitting distortion and iterative accumulation errors, and achieving high-precision and high-efficiency calculus solutions under all operating conditions.

[0042] This invention preferably adopts a finite-order orthogonal series expansion method with full-order symmetry truncation of [-M~M] as the core solution rule, covering all orthogonal series expansion forms with orthogonal completeness in multi-dimensional solution scenarios, and is not limited to a single system; this embodiment preferably adopts the Fourier series orthogonal expansion method as a typical implementation proof scheme, which is technically mature, has strong optical path adaptability, and can effectively improve the engineering implementation and stability of the scheme.

[0043] 2.2 Overview of Core Technology Solutions This invention relies on a top-level architecture of controlled laser field isomorphic computation to build a dedicated optical field calculus solution system. Its core is to perform standardized modeling of various engineering solvable functions using orthogonal series expansions. It utilizes the parallel operation rules of vector inner product from a prior patented vector numerical optical computation to solve for finite-order orthogonal series coefficients, ensuring modeling accuracy and completeness. This overcomes the industry shortcomings of existing optical field calculus solutions, such as limited adaptability, scenario limitations, and fragmented computational architectures. The core technical solution can be summarized as follows: This invention constructs seven underlying functional units, strictly defines the adaptation boundaries of engineering functions, and completes hierarchical regularization modeling of various compliant engineering functions based on a three-layer differentiated adaptation mechanism; it innovatively employs a core mechanism of single-dimensional control and multi-dimensional steady-state maintenance and independent modal evolution, unifying the univariate to arbitrary finite multi-dimensional calculus operation paradigm, supporting free switching between differential and integral operators; it conforms to the physical operation boundaries of the optical field, employing differentiated amplitude-phase modulation physical logic for differential and integral operations, and supplementing missing integral terms through precise closed-loop electronic computation, forming a photoelectric collaborative division of labor computational architecture; and it is configured with dual modes. The independent physical isolation architecture completely avoids modal crosstalk and optical field distortion caused by multimodal superposition; it adopts a finite-order harmonic symmetric truncation modeling paradigm to flexibly balance solution accuracy and computational efficiency; it reuses the optical field parallel computing mechanism to achieve integrated calculus computation through independent differential mode amplitude and phase modulation; it is equipped with a dual-time-series operation mechanism of initialization modeling and normalized constant operation, and relies on the controlled evolution of a single global optical field to generate an analytical optical field and achieve accurate direct reading at any point, ultimately achieving a stable calculus solution with high accuracy, low latency, and no cumulative error under multiple working conditions, multiple dimensions, and multiple orders.

[0044] 3. Technical Solution This application presents a novel and original independent technical solution for numerical computation of laser field calculus. Building upon the applicant's prior application of a four-step framework for isomorphic computation of physical fields, it iteratively optimizes and innovates engineering applications for laser physical carriers and specific engineering function calculus solution scenarios, forming a unique four-step closed-loop engineering implementation system. This system sequentially completes the entire closed-loop process of laser field evolution space construction, function isomorphic representation, controlled evolution computation of the optical field, and accurate result reading. This application does not claim protection for general-purpose frameworks, known hardware, known optical mechanisms, or general mathematical tools themselves; it only protects the unique overall optical computation technical solution that combines known mechanisms with the invention's proprietary top-level architecture and detailed implementation mechanisms, adapting to engineering calculus solution scenarios.

[0045] This technical solution strictly follows the four-step standardized isomorphic evolution process outlined in the prior patent application. It features a unique closed-loop operating mechanism and modular decoupling architecture specific to calculus, and completes dedicated adaptation and extension for calculus scenarios based on unified top-level rules. The core innovation of this invention lies in breaking down the barriers of traditional frequency domain computation architectures, customizing and adapting known mathematical tools to laser field calculus solution scenarios, and creating a unique optical field modeling architecture, controlled evolution mechanism, and closed-loop calculus solution system. It abandons the traditional redundant "forward transformation-operation-inverse transformation" link, focusing on global isomorphic modeling and directly controlled evolution of the optical field, eliminating cross-domain redundant errors and information loss. The out-of-domain mapping and inverse transformation restoration of a very small number of special functions serve as auxiliary fallback methods, do not participate in the core computation link, do not change the core innovative boundaries of this invention, and ensure logical consistency and stable operation of the solution.

[0046] This technical solution follows a progressive architecture of "univariate basic system → multidimensional extended application". It first constructs a complete closed loop for solving univariate function calculus, and then, based on the same mathematical rules and differentiated engineering mechanisms, it extends to adapt to scenarios of multivariate partial differentials and multiple iterated integrals of arbitrary D-dimensional multivariate functions. The complete four-step standardized solution process is as follows: 3.1 Solution Procedure for Calculus of a Single-Variable Function 3.1.1 Constructing a laser field state evolution space adapted for calculus calculation

[0047] This step is the global parameter and physical field initialization phase, corresponding to the first step of the previously applied physical field isomorphic calculation. The core is to complete the underlying capability building of the controlled laser field, establishing a standardized, adaptable system of basic elements for laser field state evolution that can be solved using univariate function calculus. This unifies the global operating parameters and physical field benchmarks, providing a complete field and modulation foundation for subsequent function modeling, encoding mapping, evolution calculations, and result reading. This phase requires the complete construction of seven underlying functional units: an orthogonal series adaptation system, a standardized fundamental frequency mode set, a unified amplitude and phase physical benchmark, a dedicated calculus operator system, an independent optical path modulation channel, an optical field encoding mapping unit, and a high-precision optical field reading unit. It also completes the core supporting elements for optical field encoding. The specific construction content is as follows: 1. Standardized orthogonal series adaptation system and fundamental frequency mode set construction: Construct a standardized orthogonal series adaptation system that is compatible with the complete orthogonal series of the whole domain and can be bidirectionally adaptively matched with the periodic characteristics of the function to be solved and the orthogonal series expansion system. Establish finite order harmonic truncation modeling rules and construct an orthogonal fundamental frequency mode set with controllable order and ordered frequency points.

[0048] 2. Unified calibration of the physical reference for amplitude and phase across the entire domain: Pre-amplitude and phase reference calibration is performed on all single-dimensional orthogonal fundamental frequency optical paths to unify the standardized physical reference for amplitude normalization and phase zeroing across the entire domain, thereby eliminating the original deviation and phase shift error of the optical path.

[0049] 3. Independent optical path modulation channel configuration: Configure multiple independent and controllable single-dimensional fundamental frequency optical path modulation channels to ensure that each fundamental frequency mode can be independently modulated, independently evolved, and without crosstalk coupling.

[0050] 4. Dedicated Calculus Operator System Construction: Build a dedicated calculus operator system that is adapted to the differential of any order and the integral of any number of multiples of a univariate function, and clarify the underlying operating rules of modal independent modulation, standardized phase shift, and amplitude scaling.

[0051] 5. Standardized optical field coding mapping unit construction: Establish the exclusive amplitude and phase joint coding operation rules for univariate scenes, clarify the standardized implementation mechanism of absolute value amplitude mapping of series coefficients and positive and negative polarity phase loading of coefficients, and establish a unique correspondence mapping relationship between orthogonal series coefficients and physical parameters of laser optical path amplitude and phase, providing standardized underlying support for subsequent function isomorphic coding modeling.

[0052] 6. High-precision global optical field reading unit construction: An integrated parameter acquisition and result analysis unit adapted to univariate scenarios for steady-state laser optical field analysis is constructed. The core unit reads two types of physical parameters: spatial field strength amplitude and spatial phase offset, from the target optical path at each fundamental frequency. It accurately acquires the corresponding parameters of the target coordinate points for each independent fundamental frequency optical path, determines the mode polarity based on the optical path phase characteristics, and obtains the single-mode calculation results by combining the amplitude parameters. The complete calculation result for the target point is obtained by accumulating and synthesizing the global mode calculation results. This unit supports complete analysis of the global continuous optical field and accurate value acquisition of arbitrary coordinates without interpolation. This matches the core characteristics of this invention: single-time global optical field evolution and direct analytical reading, directly reading the original continuous steady-state physical characteristics of the optical field without the need for discrete sampling or interpolation fitting approximation processing.

[0053] 7. Construction of Physical Isolation Unit for Optical Field Crosstalk Prevention: A standardized physical isolation infrastructure adapted to single-dimensional multi-modal superposition conditions is constructed. A dual-mode protection bottom-layer unit is configured with modal-specific isolation and full-domain full-coverage isolation. The two types of isolation modes can be selected and flexibly adapted as needed according to the computing conditions, effectively avoiding crosstalk of multi-order fundamental frequency modes and optical field aliasing distortion, and ensuring the stability and accuracy of the entire process of optical field modulation, evolution and computing.

[0054] By constructing the aforementioned seven underlying functional units, a complete underlying hardware and mathematical infrastructure is formed, encompassing laser field function modeling, encoding mapping, physical computation, signal reading, and result analysis. After completing the construction and benchmark standardization of all the aforementioned basic elements, this step further develops closed-loop field capabilities, specifically including the following three fixed core capabilities to support the entire calculus solution process.

[0055] First, it constructs a standardized orthogonal series system and the ability to express function isomorphisms. This invention is compatible with all orthogonal series expansion methods that possess orthogonality and completeness. Any adaptable orthogonal series system must adhere to its own inherent mathematical constraints, satisfying the rules of orthogonality, completeness, and ordered frequency domain distribution. The series normalization constraints are incorporated into the expansion coefficients, forming a standardized finite M-order fundamental frequency set with controllable order from low to high frequency. For natively adaptable functions, the system's orthogonal series and the target function's inherent wave function are homologous, allowing for direct and accurate modeling without additional normalization. In the univariate function scenario, this invention uses a combined amplitude and phase coding system to complete the physical mapping of the optical field. The absolute values ​​of the series coefficients are loaded through optical path amplitude modulation, and the positive and negative polarity information of the series coefficients is directly loaded into the optical path based on the differential modulation of the reference phase and the π-offset phase, completely covering all the mathematical information of the series coefficients. Polarity determination is autonomously completed based on the phase evolution extremum characteristics of the light field itself, without the assistance of external algebraic identifiers. After the evolution is completed, the calculus calculation results with positive and negative signs can be directly output, forming a simple closed-loop physical mapping mechanism. The [-M~M] symmetric full-order finite truncation system is preferred, and it can be adapted to other truncation forms as needed.

[0056] Second, it constructs the capability for independent modulation and regularized controlled evolution of calculus operators. Each fundamental frequency component can be independently matched with a dedicated calculus operator, and the scaling mechanism is consistent throughout: the differential operation uses the nth power value of the corresponding fundamental frequency as the scaling factor, achieving a proportional amplification of the optical path amplitude of the corresponding fundamental frequency, while simultaneously matching the normalization factor n corresponding to the order of operation n. Positive phase shift, together completing optical field modulation; Integral operation: When the target dimension has a non-zero frequency, the reciprocal of the nth power of the corresponding fundamental frequency is used as the scaling factor to achieve inverse proportional attenuation of the optical path amplitude of the corresponding fundamental frequency, while matching the normalization n times corresponding to the operation order n. Reverse phase shifting, together completing optical field modulation, achieves controlled evolution of the full-order optical field according to predetermined rules; when containing the target dimension 0 frequency, according to The amplitude of the optical path is adjusted by a factor while the phase remains unchanged. Operator modulation is the core mechanism for optical field evolution. A single differential / integral operation corresponds to a single action of the basic operator. The nth differential / nth integral can be iterated n times by the basic operator (where the target dimension is present at 0 frequency, the amplitude is multiplied by a factor in the nth iteration). (without phase invariance), or a pre-synthesized n-order global operator that acts once, the two methods are mathematically equivalent and can be adapted to different engineering application scenarios, providing core execution support for calculus evolution operations.

[0057] Third, it establishes a high-precision optical field feature reading capability. It can accurately read the global field value and phase information after the optical field evolves, and supports the accurate reading of the calculation results at any target coordinate point, ensuring the integrity and accuracy of the output results.

[0058] The above three capabilities constitute a complete closed-loop field system, supporting the entire process of calculus solution. 3.1.2 Isomorphic expression of the laser physical field for the differential function to be solved

[0059] This stage utilizes differentiated, well-known preprocessing methods to regularize various compliant engineering functions into standard forms of finite-order orthogonal series. Relying on standardized physical representation processes such as dedicated optical path coding, amplitude-phase modulation, and mode superposition, it achieves accurate and distortion-free simultaneous modeling of the unary function to be solved into the laser physical field. It clarifies the complete physical representation practical process, strictly limits the engineering solution boundary, and completely fills the technical gap of traditional techniques that only model without specific physical representation methods.

[0060] The univariate functions within the applicable range of this invention, after preprocessing and regularization, are preferably represented as finite-order orthogonal series in the interval [-M to M]. The general standard formula is as follows: The exact formula for calculating the coefficients of the corresponding orthogonal series expansion is as follows: In the formula, T is the standard period of the regularized univariate function. For the i-th order orthogonal fundamental frequency, The function to be solved for standard periodicity is the preprocessed and regularized function. M is a controllable finite harmonic cutoff order, which can be flexibly configured according to the function smoothness, solution order, and engineering accuracy, taking into account both convergence accuracy and computational efficiency. It is an orthogonal fundamental frequency wave function basis. It is the inverse fundamental frequency wave function. The coefficients are used in the series expansion. This formula allows for the precise calculation of the series expansion coefficients corresponding to each harmonic, providing accurate mathematical parameters for subsequent optical field physical mapping. The core innovation of this invention lies in defining a practical and standardized method for representing the physical isomorphism of laser optical fields. The entire process relies on a five-step proprietary process: "function regularization - coefficient solving - optical path amplitude and phase joint encoding - single-mode independent optical field generation - superposition and reproduction of global single-optical-path calculation results." This process achieves accurate representation of the univariate function laser physical field. The complete process of the representation method is as follows: Step 1: Function Standardization and Preprocessing. Based on the three-layer function adaptation system of this invention, adaptation processing is performed on native matching functions, conventional normalizable functions, and out-of-domain mapping fallback functions. Through periodic normalization, scale substitution, interval truncation, and cross-domain mapping, any compliant univariate engineering function is uniformly normalized into a standard periodic function form adapted to a symmetric finite-order orthogonal series in the range [-M to M]. This eliminates modeling adaptation biases caused by function form, period, and domain, and satisfies the prerequisite for simultaneous modeling of the laser light field.

[0061] Step 2: Global Discrete Sampling and Precise Solution of Series Coefficients. For the normalized standard periodic function, high-density, full-coverage discrete uniform sampling is performed within a single-period spatial domain to collect sufficient spatial function value samples, fully covering the function's global fluctuation characteristics. Based on the applicant's prior patent's vector numerical parallel optical computation rules, and combined with the inherent periodic parameters of the univariate function to be solved, global parallel orthogonal inner product operations are performed on all sampled data, sampling step sizes, and the values ​​of the inverse fundamental frequency wave function group corresponding to each sampling point. This allows for a one-time, precise, and global solution to obtain the orthogonal series coefficients corresponding to all orders. It accurately quantifies the amplitude weights of each harmonic mode, providing precise mathematical parameters for optical field physical mapping.

[0062] Step 3: Physical Mapping of Amplitude-Phase Joint Encoding for Univariate Scenes. This stage uniformly adopts the amplitude-phase joint encoding rule for univariate function scenes, which is the core step in representing the function's optical field. The specific encoding logic is as follows: the amplitude of the corresponding i-th order orthogonal fundamental frequency laser path is modulated to the obtained series coefficients. The absolute value is used to complete the physical matching of the amplitude dimension; at the same time, relying on the one-to-one correspondence between the preset extreme phase and the positive and negative signs of the coefficients, the corresponding laser optical path phase is modulated with the corresponding series coefficients. The sign of the series coefficients is used to achieve a complete physical mapping of all mathematical information (amplitude + polarity), and to establish a unique correspondence between the mathematical function and the amplitude and phase parameters of the laser field.

[0063] Step 4: Independent optical field generation for single-order fundamental frequency modes. Based on the calibrated standardized fundamental frequency optical path, each orthogonal fundamental frequency laser is independently driven to generate a corresponding steady-state optical field according to the amplitude and phase parameters encoded for each mode. Each optical path corresponds to a single-order harmonic mode. The optical fields of each mode are pairwise orthogonal, independently distributed, and free from initial crosstalk. The single-path optical field accurately reproduces the spatial distribution characteristics of the corresponding harmonic.

[0064] Step 5: Global optical field shaping and functional isomorphic representation. Standardized amplitude-phase joint encoding is uniformly performed on all DC terms and AC harmonic mode optical paths within the [-M~M] interval. After all optical paths complete the full amplitude-phase joint encoding, the resulting global laser optical field is the complete and equivalent physical representation of the normalized original function. The global distribution shape, amplitude, and phase characteristics of the optical field perfectly match the original normalized function, accurately completing the physical isomorphic modeling and reproduction of the univariate function in the laser optical field.

[0065] Based on the above standardized representation method, the univariate fundamental frequency optical field has a unique steady-state distribution characteristic. It can autonomously complete unambiguous polarity discrimination within a single optical path by relying on the extreme value characteristics of optical field phase shift. It can directly realize the physical mapping and subsequent output of the signed operation results. There is no information loss or morphological distortion throughout the process, and the modeling accuracy fully matches the engineering truncation accuracy requirements.

[0066] The function adaptation and corresponding representation logic are clearly layered, adapting to the differentiated representation requirements of different types of engineering functions: native matching functions require no preprocessing and directly execute the order-corresponding fundamental frequency-photon joint encoding to complete accurate optical field representation; regular periodic / aperiodic functions are normalized and adapted, and then a standardized five-step representation process is executed; special complex functions are normalized by out-of-domain mapping and uniformly adapted to the optical field representation process of this invention. This invention explicitly excludes function scenarios that cannot converge with finite-order truncation or cannot be orthogonally expanded, as these functions lack effective optical field isomorphic representation conditions. At the same time, relying on the inherent characteristic of rapid decay of high-frequency coefficients of smooth engineering functions, high-order operations do not require explosive increases in the truncation order, thus stably ensuring the accuracy of optical field representation and subsequent calculations.

[0067] In summary, this stage uses the complete and practical five-step physical representation method to accurately transform abstract mathematical functions into concrete, modulated, and evolvable steady-state physical light fields of lasers. All normalized equivalent standard periodic functions can be standardized orthogonally modeled and isomorphically reproduced in the light field through this unified process, providing accurate and stable physical model support for subsequent controlled evolution of the light field and calculus equivalent calculations. 3.1.3 Equivalent Calculus Numerical Calculation of Controlled Evolution of Laser Physical Field

[0068] This stage implements modal-independent operator modulation for the normalized finite-order orthogonal series optical field, strictly distinguishing the differential and integral scaling mechanisms: the differential relies on the power multiple of the fundamental frequency point to achieve the optical path amplitude amplification of the corresponding fundamental frequency, and the integral relies on the reciprocal multiple of the power of the fundamental frequency point to achieve the optical path amplitude attenuation of the corresponding fundamental frequency. The standardized phase offset corresponding to the operation order n is matched respectively, and the arbitrary finite number of differential and multiple integral operations are equivalently completed, solidifying the multi-layer exclusive physical operation mechanism and forming an unavoidable underlying technical barrier.

[0069] Based on the inherent analytical properties of orthogonal series, each order of fundamental frequency component is independently matched with the corresponding calculus operator, and calculus numerical calculations are equivalently completed through the precise controlled evolution of the optical field.

[0070] one yuan The general evolutionary expression for the second-highest derivative:

[0071] one yuan The general evolutionary expression for multiple integrals: This invention strictly defines the boundaries of photoelectric operations, with clear and unambiguous division of labor: 1. Pure optical field calculations: responsible for orthogonal oscillations and DC mode series terms. The computation is completed through parallel physical evolution of the light field, with normalized timing. No electronic computing power overhead; 2. Pure electronic completion calculation: nested summation terms for multi-order boundary compensation It employs precise closed-form numerical computation without iteration or fitting, resulting in no cumulative error or fitting distortion.

[0072] This step relies on the mechanism of differentiated modulation and photoelectric synergistic term completion to achieve accurate integral solutions. The solution accuracy and applicability are precisely controllable by the finite truncation order M. As can be seen from the above general evolutionary expression of calculus, the univariate DC term can also be excluded from the optical field evolution. The DC term does not need to be considered in the differential calculation, and the DC term can also be included in the electronic completion calculation in the integral calculation.

[0073] In summary, this invention, based on the modal independent evolution mechanism, constructs a high-precision, differentiated univariate function optical field calculus solution system, clearly distinguishing the rules for differential and integral operations. Differentiation relies on the power-law amplification of the AC fundamental frequency mode amplitude and positive phase shift, achieving arbitrary-order analytical computation in a purely parallel optical field manner, resulting in a simple link and strong real-time performance. Integration employs a photoelectric collaborative architecture with optical field evolution as the primary method and electronic closed-loop non-iterative error correction as a secondary method. This solution abandons the traditional three-segment redundant computation link, achieving switchable calculus operators, unified high- and low-order computation paradigms, and globally constant optical field evolution rules. It fundamentally eliminates the industry pain points of traditional numerical calculus, such as explosive computing power, insufficient accuracy, and poor stability, from a physical mechanism perspective. 3.1.4 Reading the Target State from the Laser Physics Field: Calculus Calculation Results

[0074] After the controlled evolution of the light field, all coordinate points in the entire domain possess precise analytical field values. This invention distinguishes between the different result reading logics of differential and integral operations, achieving precise numerical aggregation. Then, it uniformly completes the accurate restoration and output of the true value through three standardized output paradigms. At the same time, it relies on a dual-time-series mechanism to ensure efficient and stable operation throughout the entire process. The specific reading logic is as follows: For univariate arbitrary-order differential operations, there is no need for electronic term supplementation. It is only necessary to directly read the global analytical values ​​in the laser light field after controlled evolution to obtain the complete differential operation result. Specifically: single-path parameter reading → internal polarity determination → superposition of global optical path results. For univariate arbitrary-multiplicity integral operations, there are boundary nested compensation remainders that the light field cannot solve. It is necessary to first read the operation results of the light field obtained from the light field evolution, and then superimpose the compensation remainder values ​​obtained by electronic closed-loop non-iterative calculation to obtain the complete integral operation values. Specifically: single-path parameter reading → internal polarity determination → superposition of global optical path results → electronic closed-loop term supplementation. After completing the aggregation of global operation values ​​for differentiation and integration, the corresponding standardized output paradigm is matched according to different function adaptation scenarios to complete the truth value restoration output.

[0075] Category 1: Uncorrected direct-read output. Adapted to native exact matching and inherent periodic exact matching functions. After modeling, this type of function has no scale distortion or morphological modification. Within finite engineering precision, it is isomorphic to the original function across the entire domain. After evolution, the field value output result can be directly read without any correction or transformation.

[0076] The second category: Pure algebraic scale correction output. This type of function adapts to heteroperiodic normalization and aperiodic equivalent adaptation functions. The preprocessing stage of this type of function involves linear substitution of coordinate scales, resulting in a scaling bias in the equivalent coordinate system calculation results. The original true value can be restored using a unified algebraic correction formula. This process requires no decoding or cross-domain transformation, and the correction process is simple and efficient. In the formula, For the original truth solution, The result is from the equivalent coordinate system calculation. As the system's reference period, The original / equivalent normalized period of the function; the differential operation takes the positive power and the integral operation takes the negative power, fully covering all scale distortion scenarios.

[0077] The third category: On-demand inverse transformation decoding and restoration. Adapted to a very small number of special fallback functions, truth value restoration is completed by relying on well-known out-of-domain inverse mapping technology. This auxiliary method does not participate in the core computation chain and only serves as a fallback guarantee for full-scenario adaptation.

[0078] Simultaneously, this invention strictly adheres to a dual-time-sequence operation mechanism: initialization modeling, coefficient solving, and function table output are all one-time fixed preprocessing overheads; the normalized operation time sequence is differentiated: the normalized time sequence for univariate differential is... No increase in computing power; univariate integral light field main operation timing Only the electronic compensation part has limited computing power overhead, and the overall timing is... This completely eliminates the shortcomings of computational power explosion and error accumulation in traditional electronic integral grid iteration.

[0079] 3.2 Extension of the univariate system to finite multidimensional multivariate functions 3.2.1 Mechanism for extending univariate systems to finite multidimensional multivariate functions The present invention provides a system for solving multiple partial differentials and multiple iterated integrals of finite-dimensional multivariable functions. It shares the same mathematical origin and paradigm as the system for solving calculus of univariate functions. It strictly reuses the four-step standardized closed-loop solution process. It only makes targeted adaptations and optimizations for the unique characteristics of multidimensional coupling, independent operation of multiple variables, and lack of autonomous criteria for polarity discrimination. There are no changes to the underlying mechanism.

[0080] This invention presents a system for solving multiple partial differentials and multiple iterated integrals of finite D-dimensional multivariable functions. The underlying mathematical principles and four-step closed-loop process paradigm of this system are consistent with those of the system for solving calculus of univariate functions. It generally follows the standardized solution loop of "field construction - function isomorphism - controlled evolution - truth reading," with targeted adaptations and extensions only for the unique characteristics of multidimensional operations. The core differences between the two are organized into five identical, non-overlapping, and fully covered exclusive differences, as follows: First, the modal composition differs: unary operations directly use independent fundamental frequency modes that are orthogonal to each other within a single dimension to participate in the operation; multidimensional operations require coupling the independent fundamental frequencies of each dimension across the entire domain to generate secondary fundamental frequency composite modes. After coupling, the original single-dimensional fundamental frequency exits the operation link, and only the composite secondary fundamental frequency modes participate in the operation, and there is no forced orthogonality constraint for cross-dimensional fundamental frequencies.

[0081] Secondly, the coding and mapping mechanisms differ: the unidimensional scenario is compatible with amplitude coding and amplitude-phase joint coding, and the polarity of the coefficients can be directly loaded through optical path phase modulation to fully realize the physical mapping of series coefficients; the multidimensional scenario is constrained by multidimensional phase coupling, and pure amplitude coding is preferred to map the absolute value of the series coefficients to the optical path amplitude. The positive and negative values ​​of the coefficients are used as independent algebraic identifiers on the periphery and do not participate in the optical path physical coding. The two have original architectural differences in physical mapping system and hierarchy.

[0082] Third, the evolution control mechanisms differ: univariate operations are based on a single-variable modal independent evolution architecture, where only the corresponding frequency points of each order are matched with dedicated operators for independent and controlled evolution, without a unified action mechanism across the entire domain and all modes; multidimensional operations adopt a dedicated control architecture of "single-dimensional directional evolution and multidimensional steady-state maintenance", which can independently apply calculus operators to the dimension of the target solution, while the modal parameters of other dimensions are locked throughout the process, accurately matching the mathematical definitions of multivariate partial differential and partial integral.

[0083] Fourth, the difference in the integration operation architecture: the photoelectric complementary term structure of univariate integral is simple, without multidimensional nested compensation structure, and the form of the remainder term is simple; multidimensional multi-iterative integral has multidimensional nested boundary compensation terms, and the remainder term architecture of electronic closed-loop complementation is more complex. The two have the same photoelectric collaborative division of labor paradigm, and only the complexity of the integral compensation structure and the nesting level are different.

[0084] Fifth, differences in polarity discrimination mechanisms: First, the truth output, scale correction, and hierarchical adaptation paradigm of multidimensional operations are completely unified, with the core difference only reflected in the polarity discrimination process; the unidimensional fundamental frequency optical field has a unique steady-state extreme value distribution and can autonomously complete unambiguous polarity discrimination; the multidimensional coupled sub-fundamental frequency optical field naturally has ambiguity in direct polarity reading due to the superposition of multidimensional phases masking extreme value characteristics, and needs to rely on known optical methods for bottom-line correction.

[0085] Based on the above five regularization architecture differences, this invention, while ensuring the common origin of underlying mathematical principles, enables the extension of the univariate system to any finite D-dimensional multivariate calculus scenario, and accurately adapts to the special computational characteristics of multidimensional systems.

[0086] The multivariate function series expansion strictly follows the univariate normalization paradigm, employing a modeling approach of "single column of pure DC terms and multivariate symmetrical coupling and summation of AC terms." This approach is consistent with the univariate system in scope and logical coherence, providing a complete mathematical foundation for multivariate four-step normalization operations. The specific expansion form is as follows:

[0087] The three-layer logic of pre-regulation of multidimensional functions, hierarchical adaptation mechanism that fully reuses the precise matching and adaptation of the univariate system, conventional preprocessing regulation, and out-of-domain mapping fallback adaptation can uniformly regulate various D-dimensional multivariate functions into the standard form of finite-differential multidimensional series. The complete expression of the multidimensional coupled series coefficients is as follows:

[0088] In the formula Let be the reference period or equivalent normalization period corresponding to the d-th dimension variable. The coefficients are uniquely solved through multidimensional global orthogonal inner product operations, precisely matching the coupling construction form of the multidimensional orthogonal series basis. In a multidimensional operational system with a single-dimensional truncation order of M and a total dimension of D, the order index of each dimension ranges from [-M, M], and the total number of coupling coefficients that can be generated globally is... It includes one set of pure DC zero-frequency coefficients and all remaining AC coupled mode coefficients.

[0089] 3.2.2 Constructing a laser field state evolution space adapted for multidimensional calculus calculations This step involves global parameter configuration and laser physical field initialization for multidimensional multivariable function calculus operations. Aligning with the initialization process of a univariate function solution system, it completes multidimensional adaptation and expansion based on a one-dimensional basic optical path architecture, building a standardized laser field state evolution space adaptable to arbitrarily finite D-dimensional multivariable function calculus solutions. This core expansion completes the construction of seven underlying functional units, comprehensively building a complete underlying capability for multidimensional optical field modeling, modulation, evolution, readout, and anti-interference protection. It unifies the physical benchmark and operating rules for multidimensional operations, laying a solid standardized physical foundation for subsequent multidimensional function isomorphic representation, directional optical field evolution, and high-precision calculus solutions. The specific construction content is as follows:

[0090] For multi-dimensional computation and multi-dimensional independent fundamental frequency optical path architectures, a global standardized amplitude and phase reference calibration is implemented for the dedicated orthogonal fundamental frequency optical path of each solution dimension. This unifies the initial amplitude of the fundamental frequency optical field across all dimensions to a unit amplitude and the initial phase to zero, completely eliminating inherent hardware deviations such as native amplitude differences and initial phase shifts between different dimensions. Through this global unified reference calibration, standardized, consistent, and traceable control of the initial operating state of the laser optical field across all dimensions and modes is achieved. This provides a unified and standardized underlying physical reference for subsequent multi-dimensional fundamental frequency global coupling to generate sub-fundamental frequency composite modes, multi-dimensional modal differentiated amplitude and phase modulation, single-dimensional directional controlled evolution, and high-precision computation result output.

[0091] For finite D-dimensional multivariate computation scenarios, a dedicated orthogonal fundamental frequency group is independently configured for each solution dimension. Within a single dimension, the fundamental frequencies of each order are pairwise orthogonal, with frequency points arranged in an ordered gradient. Furthermore, the frequency point distribution of the fundamental frequencies in each dimension can be bidirectionally adaptively matched with the periodic characteristics and series expansion forms of the corresponding dimension variables. After completing the amplitude and phase benchmark calibration of the single-dimensional fundamental frequencies in each dimension, all independent orthogonal fundamental frequencies are coupled globally to generate a multidimensional secondary fundamental frequency composite mode that accurately matches the multivariate function series basis construction. After the mode coupling is completed, the original single-dimensional fundamental frequencies automatically exit the multidimensional computation link, retaining only the secondary fundamental frequency composite mode as the sole valid computational unit throughout the entire process, with no invalid single-dimensional mode residues.

[0092] The configuration of multi-dimensional independent optical path modulation channels means that the objects of multi-dimensional calculus optical path calculation are each secondary fundamental frequency optical path mode. Each secondary fundamental frequency mode optical path is provided with a dedicated optical path modulation channel with independently adjustable parameters. The optical paths are not coupled to each other and do not interfere with each other.

[0093] For the unique sub-fundamental mode object in the evolutionary operation stage, an independent, dimension-specific calculus operator system is constructed. A single optical path uniquely corresponds to a multi-dimensional coupled sub-fundamental mode, precisely matching the core mechanism of this invention: applying calculus operators to the target solution dimension optical path to complete directional evolution, while the parameters of other non-target dimension optical paths are locked and maintained in a steady state throughout the process, strictly conforming to the mathematical definition of "single-variable solution, constant other variables" in multivariate partial differential equations and multiple accumulated integrals. The operator control rules are mathematically and logically consistent with the univariate solution system.

[0094] Adapting to the characteristics of multidimensional multivariate function coupled modeling, a dedicated standardized optical field encoding and mapping unit is built. A pure amplitude encoding format is uniformly adopted across the entire domain to construct a stable mapping relationship between multidimensional coupled series coefficients and laser optical field physical parameters. During the encoding process, only the absolute values ​​of the multidimensional coupled series coefficients are mapped to the corresponding sub-fundamental frequency optical path amplitude, achieving precise amplitude dimension matching. The positive and negative polarities of the series coefficients are uniformly set as independent algebraic identifiers on the periphery, not participating in optical path phase modulation and physical encoding, effectively avoiding polarity encoding ambiguities caused by multidimensional phase superposition. This encoding rule is adaptable to any finite-dimensional, multimodal coupled optical field modeling scenario, providing standardized underlying encoding support for the simultaneous construction of modes and independent modal evolution operations of multidimensional function optical fields.

[0095] A steady-state analytical optical field acquisition and result analysis unit adapted to the characteristics of multidimensional composite optical fields is constructed. This unit accurately acquires the core physical parameters of multidimensional superimposed phase shifts, independently identifies the evolution state of each composite mode, and accurately aggregates the single-mode computation results. Addressing the inherent characteristic of multidimensional coupled optical fields lacking autonomous polarity criteria, a dedicated polarity correction fallback mechanism is implemented, combining algebraic identifiers of peripheral coefficients to achieve accurate polarity correction across the entire computation domain. This unit abandons the traditional reading paradigm of discrete sampling, point-by-point fitting, and iterative approximation. It supports complete analysis of multidimensional global continuous optical fields and accurate value acquisition at arbitrary spatial coordinates without interpolation. Relying on a single global optical field evolution, it can complete the aggregation of multidimensional, multimodal global computation results, outputting a complete analytical solution and ensuring the integrity, accuracy, and stability of high-dimensional computation results.

[0096] To address the unique challenges of cross-dimensional coupling, multimodal aliasing, and sub-fundamental frequency crosstalk distortion inherent in multidimensional computation, a standardized dual-mode physical isolation underlying protection unit is constructed to comprehensively ensure the stability of the entire process of multidimensional optical field modulation, evolution, and computation. The unit comprises two types of independently deployable, on-demand standardized isolation architectures: First, targeted physical isolation, which precisely locates high-risk interference units such as high-order single-dimensional fundamental frequencies and easily distorted sub-fundamental frequency composite modes, implementing precise point-to-point isolation, adapting to low-to-medium complexity multidimensional computation scenarios, balancing isolation accuracy and deployment cost; second, full-domain, full-coverage physical isolation, which uniformly isolates all fundamental frequency components and all sub-fundamental frequency composite modes across the entire domain, completely eliminating cross-dimensional crosstalk, mode aliasing, and optical field distortion problems under high-dimensional, high-order, and multimodal superposition conditions, adapting to high-precision, high-stability complex computation scenarios. Both isolation modes can be flexibly adapted according to the solution dimension, truncation order, and number of modes, building a solid barrier for the stability of multidimensional computation.

[0097] Based on the complete construction of the above seven underlying functional units, this step ultimately forms a multi-dimensional closed-loop core computing capability, encompassing three core functions: standardized isomorphic representation of multi-dimensional engineering functions, independent and controlled evolution of multi-dimensional sub-fundamental frequency modes, and direct analytical reading of any point in the multi-dimensional global domain. Coupled with a dual-mode physical isolation protection architecture, it comprehensively avoids multi-modal coupling crosstalk and computational distortion problems. It should be noted that the description of this invention—first constructing a single-dimensional fundamental frequency wave function and then coupling to generate a multi-dimensional sub-fundamental frequency wave function—is merely for clarity of step-by-step logic and ease of technical explanation, and does not constitute a limitation on the actual engineering implementation. The engineering side can implement this by directly generating the sub-fundamental frequency wave function mathematically, equivalently realizing the multi-dimensional modeling and computation mechanism of this invention.

[0098] 3.2.3 Isomorphic Expression of Laser Physical Fields as Multidimensional Multivariable Functions By adopting a dedicated hierarchical function adaptation and standardized modeling logic, various compliant D-dimensional multivariate engineering functions are regularized into a standard expansion form of a multidimensional finite-order fundamental frequency wave function group, realizing a precise isomorphic physical mapping of multivariate functions to multidimensional laser light fields.

[0099] First, hierarchical preprocessing and regularization of multidimensional functions are performed. For various types of original multidimensional functions, morphological regularization is achieved through a three-layer adaptation mechanism: precise matching and adaptation, conventional preprocessing regularization, and out-of-domain mapping as a fallback. Original matching functions directly complete multidimensional series expansion; periodic and aperiodic functions are equivalently regularized through conventional and well-known methods such as multidimensional scale substitution, periodic normalization, and effective interval truncation; a very small number of complex multidimensional coupled functions rely on out-of-domain mapping to complete the preliminary morphological adaptation, and are finally uniformly transformed into standardized multidimensional periodic equivalent functions that satisfy the finite-order truncation expansion condition.

[0100] Secondly, the multidimensional coupling coefficients are solved and standardized modeling is completed. The normalized multidimensional standard function is subjected to full-domain discrete sampling, and the patented parallel computing logic of vector numerical light computation is reused to solve all multidimensional series coupling coefficients in parallel across the entire domain in one go, without iteration error or serial operation deviation, thus completing the standardized series modeling of the multidimensional function.

[0101] Finally, the physical representation of the multidimensional optical field amplitude encoding is completed. The physical mapping of the optical field is completed using a pure amplitude encoding rule. Only the absolute values ​​of the multidimensional series coupling coefficients are mapped and modulated to the amplitude parameters of each fundamental frequency optical path. The positive and negative signs of the series coefficients are uniformly used as independent algebraic identifiers on the periphery and do not participate in the physical modulation and phase encoding of the optical path in this stage.

[0102] 3.2.4 Equivalent Multivariable Calculus Numerical Calculation for Controlled Evolution of Multidimensional Laser Physics Field Based on the unified operator modulation and amplitude-phase control mechanism, it adapts to the mathematical characteristics of multidimensional multivariate multiple partial differentials and multiple iterated integrals, and completes the precise controlled evolution of multidimensional optical fields, which is equivalent to realizing high-dimensional calculus numerical operations. The core follows the core rules of "dimensional independent operator binding, single-dimensional directional evolution, and multidimensional steady state maintenance".

[0103] The multidimensional operator control follows the core operational logic of "dimensionally independent operator binding, independent evolution of secondary fundamental frequencies, unidimensional directional evolution, and multidimensional steady-state maintenance." Multidimensional computation uses all coupled secondary fundamental frequency composite modes across the entire domain as a unified computational unit. Each dimensionally independent fundamental frequency is pre-bound with a dedicated calculus operator, and all secondary fundamental frequency composite modes across the entire domain participate in controlled evolution computation. During the computation, each secondary fundamental frequency composite mode across the entire domain solves for its own dimension-specific fundamental frequency component based on its coupled target, matching the corresponding fundamental frequency's dedicated frequency point operator to complete the corresponding controlled evolution.

[0104] Solve the d-th dimension of the objective function using multiple partial derivatives. Based on the corresponding base frequency point The exponential parameter is used to proportionally amplify the optical path amplitude of the second fundamental frequency mode coupling component and synchronize it. Double positive Standardized phase shift; multiple iterated partial integral operation in the d-th dimension, excluding the d-dimensional zero-frequency secondary fundamental mode, based on the corresponding fundamental frequency point. The reciprocal parameter is used to apply inverse proportional attenuation of the optical path amplitude to the coupling component of the second fundamental frequency mode, and is synchronously matched. Double reverse Normalized phase shift; multiple iterated partial integral operation in the d-th dimension, containing the d-dimensional zero-frequency secondary fundamental frequency mode, according to... The optical path amplitude is adjusted by a factor, while the phase remains unchanged. The degree of differentiation and the multiplication factor of integration are configured independently for the target dimension to be solved. When the dedicated calculus operator for the target dimension is executed, the non-target dimensions remain in a steady state throughout, strictly conforming to the mathematical definition of multivariable partial calculus.

[0105] For any The general expression for calculating D-dimensional multivariable partial differentials is as follows: General expression for multivariable partial differential calculation: ; General expression for calculating multivariate and multiple iterated integrals: .

[0106] Multidimensional computation employs a photoelectric collaborative division of labor model. Multidimensional partial differential operations are completed entirely through parallel evolution of pure optical fields, with no electronic computing power overhead and constant timing. In multidimensional multi-iteration integral operations, multidimensional coupled modal series terms... Completed by parallel physical evolution of the light field; multi-dimensional nested boundary compensation terms. By using a closed-loop, precise, non-iterative, and non-fitting electronic system to complete the calculations, the industry pain points of explosive computational power and layer-by-layer error accumulation in traditional high-dimensional integration are completely avoided. As can be seen from the above general calculation formula, the pure DC term can also be excluded from the optical field evolution, the result of multidimensional partial differential equation calculation remains unchanged, and the electronic calculation completes the calculations during multidimensional multivariate multiple iterations.

[0107] 3.2.5 Calculus Calculation Results for Reading Target State from Multidimensional Laser Physical Field Based on the three standardized output paradigms and dual-time operation mechanism preset in this invention, accurate true value restoration and stable output of multidimensional calculus calculation results are achieved.

[0108] First, the total value of the multidimensional optical field calculation and the electronic complement value are solved by combining them. Due to the inherent characteristics of the multidimensional phase superposition of multidimensional sub-fundamental frequency composite modes, the multidimensional optical field has no unique steady-state extremum to support direct polarity discrimination, and it is impossible to directly rely on the optical field itself to complete unambiguous polarity determination. After the multidimensional optical field completes controlled evolution, a globally continuous analytical multidimensional optical field is generated, and the values ​​of the optical path evolution at any coordinate point in the spatial domain can be directly read. This invention relies on known conventional technologies in the field, such as time series analysis and optical field polarization detection, to complete the basic polarity correction of the optical path. These known auxiliary technologies are only used as a backup means for multidimensional scene polarity discrimination, and do not participate in the core optical field modeling and evolution calculation link, nor do they change the core innovative boundary of this invention.

[0109] A two-layer polarity determination logic is employed. After reading the sub-fundamental frequency field value of a single optical path, the basic polarity determination is first completed using the aforementioned conventional techniques. Then, the peripheral positive and negative algebraic identifiers preset by the optical path are superimposed to obtain the single-path signed operation result. The single-path results of all modes are then summed across the entire domain to obtain the total value of the pure optical field operation. For multi-dimensional, multi-iteration integral operation scenarios, the above total value of the optical field operation needs to be merged with the closed-form precise completion of the multi-dimensional nested boundary compensation term of the electronic system across the entire domain to obtain the complete original value of the calculus operation. All subsequent truth value restoration operations are based solely on this merged complete original value.

[0110] Second, it is based on the hierarchical truth value restoration output of the complete original value. Taking the complete original value of the operation after merging the light field and electronic complement as the benchmark, it adapts to three types of engineering conditions to complete the final truth value restoration output: the original matching regularized function can be directly read and output as the merged original value; the equivalent function after scale substitution and periodic normalization is restored to multidimensional truth value through a unified pure algebraic scale correction formula based on the merged complete original value; for a very few special and complex multidimensional functions, the merged complete original value is adapted and corrected for a fallback output by relying on known inverse mapping decoding and out-of-domain restoration methods, which fully covers multidimensional engineering operation scenarios and completely avoids the mathematical deviation and result distortion problems caused by separate calculations and individual restorations. At this stage, the underlying mathematical mechanism of univariate and multidimensional calculus solutions is completely consistent and unified, only multidimensional multivariate operations have higher engineering implementation complexity due to the dimensional coupling characteristics.

[0111] Third, it achieves stable operation across multiple dimensions and dual time series. It retains the dual-time series architecture of one-time initialization preprocessing and constant normalization operations. The normalization time series for multidimensional, multivariate, multiple partial differential operations is as follows: The computational timing of the multidimensional, multivariate, and multicumulative integral optical field calculation part is also... Only the electronic compensation stage has a controllable low-order computing power overhead. The overall time series grows in a controllable polynomial manner with the dimension and degree / multiplicity. There is no iterative accumulation error or computing power explosion, which ensures efficient and stable operation of multi-dimensional high-frequency and high-precision solution conditions.

[0112] 3.3 Operator aliasing risk and standard physical isolation architecture This invention features pairwise orthogonal fundamental frequency modes with no natural crosstalk, exhibiting extremely strong modal independence in one-dimensional computation, virtually eliminating the risk of aliasing and distortion. In contrast, multi-dimensional computation lacks mandatory orthogonal constraints on the fundamental frequencies of each dimension, resulting in naturally occurring cross-dimensional and cross-modal crosstalk and aliasing defects in the coupled secondary fundamental frequency composite modes. Distortion issues are particularly prominent in multi-mode superposition scenarios. Addressing this core pain point, this invention innovatively proposes two types of independent, on-demand, and fully adaptable standardized physical isolation architectures, both falling within the scope of this invention: The first type is targeted physical isolation: accurately locate the high-order fundamental frequency and multi-dimensional coupled sub-fundamental frequency components that have the risk of crosstalk and aliasing, and accurately implement physical isolation, taking into account both isolation accuracy and engineering implementation cost; The second type is full-area, full-coverage physical isolation: all fundamental and secondary fundamental frequency modes within the field are uniformly isolated throughout the entire process, completely eliminating all crosstalk and distortion risks.

[0113] The two isolation schemes have no priority distinction, can be implemented independently, and can be flexibly selected as needed, comprehensively ensuring the independence, stability and accuracy of optical field calculations under all working conditions.

[0114] 4. Beneficial effects This invention addresses the shortcomings of existing optical field calculus techniques, such as poor adaptability, fragmented systems, susceptibility to crosstalk and instability, explosive high-dimensional computing power, and weak dimensional expansion. It leverages a unique four-step closed-loop isomorphic evolution architecture, a hierarchical function adaptation system, a single-dimensional controllable multi-dimensional steady-state mechanism, and a dual-mode physical isolation architecture to achieve comprehensive technical advantages. Specific beneficial effects are as follows: (1) The patent barrier is complete, the architecture is clear, and the adaptability is excellent. This invention is fully compatible with the four-step basic framework of general physical field isomorphic calculation applied for earlier. There is no need to change the core logic. It builds a progressive protection barrier of "general top-level framework + exclusive calculus sub-scenarios". The modules are clearly divided and the technical boundaries are complete. It completely solves the defects of traditional technical barriers that are weak and easily circumvented by fine-tuning. It can be implemented in modular and layered engineering.

[0115] (2) A seven-layer, all-dimensional solution system with wide applicability and clear boundaries. This invention features a unique seven-layer, all-dimensional adaptation system encompassing function adaptation, order solving, variable dimensions, operational architecture, operator switching, multiple unifications, and full-domain analysis and direct reading. This system overcomes the shortcomings of traditional technologies, which often have single-layer adaptation and fragmented systems. First, it offers a wide range of function adaptations, with a three-layer hierarchical adaptation mechanism covering various types of engineering functions, including periodic, non-periodic, decaying, and time-varying functions, and is compatible with all orthogonal complete series systems. Second, it provides flexible and controllable accuracy through the order M. It offers several advantages: 1) Accuracy can be configured as needed to adapt to various solution requirements; 2) It boasts strong dimensional scalability, supporting multivariate partial differential and multi-iterative integral solutions from one dimension to any high dimension, with unified underlying rules; 3) It features a unified computational system, with differential and integral operations relying on a unified mathematical modeling and evolution system for laser light fields; 4) Operators are bidirectionally switchable, with a single architecture compatible with both differential and integral operations, covering a wide range of application scenarios; 5) It unifies high- and low-order computational paradigms, completely resolving the shortcomings of fragmented traditional algorithm processes and system reconstruction; 6) It enables direct reading of global analytical evolution, relying on a single light field evolution to output continuous analytical solutions without discrete iteration errors, achieving accurate true value reading at any point.

[0116] (3) Dual-mode physical isolation architecture, which greatly improves the stability and accuracy of operation. The present invention is configured with two independent standardized isolation schemes, which can be adapted to simple, complex, low-order, high-order, one-dimensional and multi-dimensional full-condition scenarios as needed, accurately avoid modal crosstalk and amplitude and phase distortion problems, and take into account the economic efficiency of engineering implementation and the stability of ultra-precise operation, effectively filling the stability shortcomings of high-order and high-dimensional complex operation.

[0117] (4) Photoelectric differentiated collaborative operation completely eliminates the pain points of computing power explosion and error accumulation. This invention fits the physical operation boundary of the light field. Differentiation is completed in the whole light field. Integration adopts a differentiated architecture of light field main operation + electronic precision filling. The missing terms of integration are filled by closed electronic calculation without iteration and fitting. It abandons the traditional simulation fitting and grid iterative approximation method, with no fitting distortion and no accumulated error. With the dual time sequence operation mechanism, the normal differential time sequence is O(C) and the integration time sequence can be controlled to grow at a low order. It completely gets rid of the exponential computing power explosion defect of traditional electronic integration. The computing efficiency, real-time performance and stability are comprehensively superior to traditional technologies.

[0118] (5) Outstanding multidimensional adaptability, accurate and distortion-free high-dimensional computation. Relying on the exclusive mechanisms of dimension-independent optical paths, single-dimensional independent operator control, and multidimensional steady-state maintenance, while retaining the coupling characteristics of multivariate functions, it accurately matches the mathematical definitions of multivariate multiple partial differentials and multiple iterated integrals, realizing independent single-dimensional computation and constant steady-state in other dimensions. I. The multidimensional technology paradigms are of the same origin, but the implementation mechanisms are different, with no dimensional expansion distortion, coupling interference, or decoupling errors. It can support large-scale high-order and high-dimensional coupled calculus solutions, solving the core shortcomings of traditional optical computing such as poor dimensional expansion and inaccurate high-dimensional computation.

[0119] (6) High precision and efficiency with flexible balance and strong engineering adaptability. The [-M,M] symmetric finite-order truncation modeling paradigm is adopted, and the solution accuracy is flexibly controllable by M, which can be adapted to various working conditions from lightweight fast solution to ultra-precise high-order solution. Relying on the rapid decay characteristics of high-frequency coefficients of engineering smooth functions, high-order calculations do not require a surge in truncation order, ensuring stable high-precision solution and achieving the optimal balance between accuracy and efficiency.

[0120] (7) Global analytical solution and direct reading at any point: revolutionary leading in solution accuracy. Unlike the traditional numerical solution paradigm of grid-by-grid iteration and discrete fitting, this invention generates a global continuous steady-state analytical light field through a single controlled evolution of the standardized light field, outputting a complete analytical solution rather than a discrete approximation solution, completely eliminating grid truncation error, iteration accumulation error, and point interpolation distortion. After the light field evolution is completed, the accurate calculation results at any coordinate point in the spatial domain can be directly read without secondary fitting and iterative correction. Relying on this original analytical solution mechanism, high-precision true value output is achieved under all working conditions, forming a revolutionary core advantage that is different from traditional numerical calculus techniques.

[0121] (8) High hardware reuse rate and excellent engineering applicability. This invention not only has a completely identical mathematical architecture and operational paradigm for fractional and multiple integrals, but also a completely unified underlying operational architecture, light field evolution process, and modal modulation rules for arbitrary high and low order differentials and arbitrary light and heavy multiple integrals. There is no need to reconstruct the hardware architecture and algorithm logic for different operation types and different orders of multiplicity. It can flexibly adapt to various fractional and multiple integral solving conditions by simply switching back-end differentiated parameters and operator rules, which greatly improves the general reusability of hardware. At the same time, it breaks away from the traditional frequency domain redundant transformation link and takes the direct evolution operation of the whole domain light field as the core. There is no cross-domain redundant error, which can be adapted to a variety of engineering implementation carriers. The cost of large-scale implementation is low and the feasibility is strong.

[0122] (9) Multi-scenario layered output closed loop ensures accurate and controllable results. Three output paradigms are established: uncorrected direct reading, pure algebraic scale correction, and on-demand inverse transformation decoding. These paradigms cover most common, regular, and special complex function solving scenarios, forming a rigorous and self-consistent truth restoration closed loop. This ensures efficient solving under normal working conditions and makes up for the shortcomings in adapting to special complex scenarios, comprehensively meeting the high-end engineering needs of industrial simulation, dynamics solving, multi-field coupling analysis, and high-precision numerical integration.

[0123] 5. Example: Example of solving the controlled evolution of laser field using calculus based on Fourier series expansion This embodiment selects the Fourier orthogonal series expansion, which is most widely used in engineering applications, as a concrete carrier for principle derivation. It constructs a laser field calculus solution derivation process that can be theoretically replicated and used for principle verification. The entire process strictly aligns with the general core technology solutions, proprietary operating mechanisms, parameter rules, and optoelectronic division of labor architecture described earlier. Unlike the comprehensive explanation of principles, mechanisms, and general rules in the general technology solutions described earlier, this section focuses only on the adaptation configuration, proprietary parameter settings, derivation process, selection strategy, and principle verification process of this principle solution. It strictly relies on seven underlying functional units, three-layer function adaptation, modal independent evolution, single-dimensional control of multi-dimensional steady state, dual-mode physical isolation, and dual-temporal operation—seven core mechanisms—to complete the calculus solution principle derivation for univariate functions and finite D-dimensional multivariate functions. Specific example parameters, proprietary selection, and fixed derivation processes demonstrate the theoretical feasibility and mechanistic stability of the invention's architecture, exemplifying that the invention can adapt to any orthogonal series expansion system and any finite-dimensional and multiplicative calculus solution scenario. In this embodiment, the pure DC term is not included in the optical field evolution; the integration of the pure DC term is directly handled by electronic-aided calculation.

[0124] 5.1 Four-Step Implementation Process for Standardizing Calculus of Single-Variable Functions The implementation process for solving this univariate function adopts a lightweight electronic-assisted mode: the optical field hardware is dedicated to performing parallel physical evolution calculus calculations for all AC harmonic modes; the back-end electronic module accurately completes the nested compensation remainder terms and pure DC terms of the integral boundary, which cannot be directly achieved by the optical field, in a closed-loop, non-iterative manner. The differential case relies solely on the pure optical field for calculation, without electronic term completion, resulting in a clear division of labor between optoelectronics and a unified, solidified process. Result reading adopts a hierarchical architecture of independent value acquisition for each fundamental frequency and electronic-assisted full-domain accumulation: after the evolution of each fundamental frequency mode, the corresponding point calculation result is read separately, and the accumulation calculation of the summation of independent single-mode results is uniformly handled by the electronic-assisted module, simplifying the optical field calculation chain and clearly defining the division of labor.

[0125] 5.1.1 Constructing a laser field state evolution space adapted for calculus calculations (physical field initialization) This embodiment focuses on solving univariate functions, outlining the principle derivation and parameter adaptation for laser field calculus solutions. It establishes a laser physics field theory derivation model adapted to Fourier series expansions, completing the entire process design for optical path reference calibration, modal parameter configuration, and isolation mechanism deployment. This provides standardized theoretical parameters and optical path architecture support for the subsequent principle verification of function modeling, evolution calculations, and result reading in this embodiment. This section only describes the concrete configuration logic of this principle derivation and does not repeat the general initialization principles described earlier.

[0126] The specific parameter configuration for this embodiment is as follows: the standard reference period of the prototype and the period of the univariate function to be solved are set to be the same, both being... Corresponding to the matching reference frequency The frequency points of each harmonic are strictly arranged according to integer multiples of the order gradient, satisfying... This ensures that the single-dimensional modes are orthogonal, independent, and regularly arranged, adapting to the inherent orthogonality of Fourier series expansions. The prototype supports adaptive fine-tuning based on the native periodicity and fluctuation characteristics of the objective function to be solved. and The parameters enable precise adaptation and matching between the function and the optical field frequency system.

[0127] In this embodiment, the one-dimensional order index is limited to Configured with 1-channel zero-frequency DC base and symmetrical AC harmonic optical path, total The prototype features an independent orthogonal optical field channel. The entire optical path loading and evolution process of the zero-frequency DC substrate is shielded, and only all [channels / paths] are used. The optical path's harmonic modes are used in subsequent modeling and calculus calculations, reducing the computational overhead of the optical path.

[0128] This embodiment fully implements a seven-core functional unit architecture and achieves differentiated adaptation by combining the orthogonality of Fourier series: Based on integer multiple gradient frequency point layout rules, it constructs an orthogonal series system and fundamental frequency mode set adapted to Fourier expansion; it completes full-domain amplitude and phase normalization and phase zeroing calibration based on a unified optical path reference; it configures multiple independent AC harmonic optical paths to achieve single-mode independent modulation channel layout; it implements a calculus-specific operator system; it builds an amplitude and phase joint encoding optical field mapping unit based on the characteristics of univariate scenarios; it builds a high-precision optical field parameter reading unit to adapt to the full-domain analytical direct reading requirements; and it completes the anti-crosstalk protection unit layout with a dual-mode physical isolation architecture. These seven core functional units comprehensively adapt to the univariate function calculus solution scenario of this embodiment, forming a complete closed-loop underlying hardware and mathematical support system.

[0129] Based on the aforementioned standardized underlying architecture, this principle derivation model forms three sets of reusable core theoretical capabilities, fully equipped with a dual-mode physically isolated hardware design, which can be adapted to the simulation verification needs of different computing conditions as needed: First, the ability to construct a model using the Fourier series of engineering functions, enabling precise mapping and replication of the normalized engineering function and the laser optical path; second, the ability for independent and controllable evolution of a single mode, supporting the individual configuration of differential and integral operators for any order of harmonic optical path, accurately completing optical path amplitude scaling and standardized phase control; and third, the ability for high-precision point reading across the entire domain, capable of acquiring the complete optical field parameters after evolution and accurately outputting the calculus solution results for any spatial coordinate point, theoretically ensuring computational stability and accuracy.

[0130] 5.1.2 Objective Function Preprocessing and Regularization and Laser Light Field Modeling and Derivation This step is a principle modeling and derivation process specific to this embodiment. Commonly used univariate periodic and non-periodic functions in conventional engineering are selected as the principle verification objects. The standardized preprocessing theoretical means adapted to this invention are used to perform concrete regularization derivation on multiple types of objective functions to be solved, and uniformly transform them into standard periodic equivalent functions adapted to the local Fourier series modeling, which meet the theoretical derivation conditions of finite-order truncation expansion. This is different from the pure theoretical explanation of the general adaptation mechanism in the previous text, and provides a principle basis for the engineering implementation of this invention.

[0131] This embodiment strictly follows the five-step physical representation process and the three-layer differentiated function adaptation mechanism, solidifying the standardized derivation link of "function regularization - discrete sampling - parallel parameter calculation - amplitude and phase encoding - modal superposition shaping": First, relying on the three-layer adaptation logic of precise matching adaptation, conventional preprocessing regularization, and out-of-domain mapping as a fallback, the objective function undergoes periodic normalization, interval truncation, and equivalent period reconstruction preprocessing to adapt to the local orthogonal series modeling paradigm and eliminate function shape adaptation bias; Second, the regularized standard periodic function undergoes high-density discrete sampling across the entire domain to fully cover the single-period spatial characteristics and ensure the accuracy of coefficient calculation; The process involves three steps: First, relying on the prior patented parallel optical computing architecture of vector numerical optical computing, all Fourier series harmonic coefficients are solved in parallel across the entire domain. Second, for univariate scenarios, amplitude-phase joint encoding is uniformly adopted, mapping the absolute values ​​of series coefficients to optical path amplitudes. The positive and negative polarities of the loaded coefficients are modulated by the differential modulation of the reference phase and π offset phase. No external algebraic identifier is required, and the signed computation results can be output autonomously based on the characteristics of the optical field itself. Third, by shielding the DC base mode, the optical paths of multiple orthogonal AC modes evolve independently and are superimposed to accurately reproduce the physical form of the objective function, thus verifying the principle of co-construction of univariate function laser optical field.

[0132] The standard paradigm for concrete modeling of unary functions in this embodiment is as follows, and is only used for the prototype deployment and verification: This formula is only the mathematical basis for the implementation of this embodiment. The actual prototype operation completely abandons the DC optical path loading and only relies on the AC mode to complete the physical modeling, which is suitable for the extremely simple and efficient engineering implementation requirements of this machine.

[0133] 5.1.3 Concrete Modulation of Optical Field Operator and Derivation of Calculus Principles This step is the core principle derivation stage of this embodiment. Based on the aforementioned optical field theoretical model, it conducts principle simulation derivations of optical path amplitude scaling and phase shift modulation for different orders of differentiation and different multiplicity integration conditions, thus verifying the mechanism of calculus operations. This principle derivation uses the same theoretical hardware optical path architecture throughout, distinguishing between differentiation and integration operation modes only through background parameter switching, without requiring hardware architecture reconstruction. This verifies the core advantages of this invention—flexible operator switching and unified operation paradigm—from a principle perspective.

[0134] The concrete expression for the linear differential in one variable in this embodiment is as follows: The concrete operational expression for the nth-order differential in this embodiment is as follows:

[0135] The fixed logic in this differential principle derivation is as follows: under a single differential operation, the theoretical model automatically reads the fundamental frequency specific to each optical path. Using the original fundamental frequency as the amplitude amplification factor, the optical path is proportionally scaled, while simultaneously matching the forward direction. Standardized phase shift matches the physical characteristics of first-order differential; all optical paths are independently modulated and do not crosstalk each other, requiring no iterative calculations. The global differential operation mechanism can be verified by simulation of the principle of global controlled evolution of a single optical field, which is completely aligned with the core mechanism of mode independent evolution and standardized phase shift of this invention.

[0136] For the high-order differential derivation scenario, the theoretical model can calculate the nth power of the fundamental frequency of each optical path based on the preset differential order n, using it as the exclusive amplification factor, and simultaneously superimpose n times the forward amplification factor. Phase shifting enables a unified principle adaptation and derivation for arbitrary higher-order differentials. This principle derivation maintains independent evolution of each optical path, with no mode mixing or parameter coupling throughout the process. Mechanistically, it verifies that higher-order operations do not require hardware or process modifications; only the post-step parameters need to be adjusted for flexible adaptation.

[0137] The concrete implementation of the single-variable single-fold integral in this embodiment is expressed as the standard expression: This embodiment concretely demonstrates the standard expression for a single-variable n-fold integral:

[0138] This derivation of the single-integral principle uses a fixed logic: the theoretical model uses the reciprocal of the original value of each fundamental frequency as the amplitude attenuation factor to complete the inverse proportional amplitude control of the optical path, and synchronously matches the reverse... Standardized phase shifts are used to accurately match the mathematical laws of first-order integrals. At the same time, the specific optoelectronic division of labor rules for this principle derivation are strictly followed, distinguishing the theoretical boundaries of optical field operations and electronic term completion, and strictly aligning with the optoelectronic collaborative operation mechanism described above, ensuring the rigor of the operation mechanism and the standardization of the process.

[0139] In this embodiment, the fixed optoelectronic division of labor is implemented as follows: the optical field hardware is only responsible for completing the parallel physical evolution calculation of the AC mode series, with constant timing and no computational overhead throughout the process; the back-end electronic module accurately completes the integral boundary nested compensation remainder term and DC term that the optical field cannot directly realize in a closed-loop, non-iterative manner, without adopting the traditional grid iteration and fitting approximation method, thus completely avoiding the iterative accumulation error.

[0140] This implementation model employs a division of labor that simplifies computation through hardware optical paths and precisely fills in gaps with electronic modules. While ensuring mathematical completeness, it maximizes the efficiency of optical field computation and adapts to the real-time solution requirements of engineering projects.

[0141] For the scenario of n-fold high-order integral derivation, the theoretical model can be based on a preset integral number n, with the inverse of the nth power of the fundamental frequency as the attenuation factor, and combined with n times the inverse... Phase shifting completes the standardization principle derivation of integrals with arbitrary multiplicity, verifying from a mechanistic perspective that the calculation process for high and low multiplicity integrals is completely unified, without the need to reconstruct the calculation logic.

[0142] The integral operation principle derivation in this embodiment follows the above-mentioned solidified optoelectronic division of labor theory model. Only the AC mode participates in the derivation of the optical field evolution mechanism, and the remaining items are derived using unified electronic closed-loop supplementary theory. The derivation process is standardized and replicable, providing complete mechanism support for the engineering implementation of this invention.

[0143] This embodiment adopts a fixed variable upper limit iterative integral theory derivation paradigm, uses a standardized single-cycle global domain as the solution basis, fixes the interval starting point as the system zero, uniquely determines the boundary conditions, and has no free integral constant. It adapts to the definite integral solution conditions in industrial engineering from the principle level, with clear implementation boundaries and strong adaptability to operating conditions.

[0144] 5.1.4 Derivation of the Principle of Light Field State Reading and True Value Reconstruction This step is the derivation process of the output principle in this embodiment. Based on the evolved global analytical optical field theoretical model, it accurately reads the two core physical parameters corresponding to the optical field spatial coordinate points: optical field intensity and phase shift. Combined with the normalized benchmark and standardized phase shift rules, it completes the parameter analysis and conversion. The one-dimensional differential operation follows the standardized process of "single optical path parameter reading → internal polarity determination → global optical path result superposition" to complete the result collection; the one-dimensional integral operation follows the standardized process of "single optical path parameter reading → internal polarity determination → global optical path result superposition → electronic closed-loop term completion" to complete the result collection.

[0145] After the calculation results are collected, the three types of fixed output paradigms are adaptively matched according to the preprocessing type of the objective function to verify the truth value restoration mechanism. Natively adapted periodic functions that do not require regularization can directly read the field value output at any point; equivalent functions that have undergone scale substitution and periodic normalization have their truth value restoration principle deduced through built-in pure algebraic correction formulas; and for a very few special and complex functions, the result correction mechanism deduction is completed based on a preset inverse mapping decoding fallback process, fully covering the principle adaptation verification for conventional engineering conditions.

[0146] 5.1.5 Comparison of the timing sequence of a single-variable nth-degree calculus with the traditional electronic numerical calculus scheme This embodiment employs a dual-temporal theoretical operation mechanism throughout: function modeling and parameter sampling are performed as one-time preprocessing theoretical operations, while the routine calculus solution only retains the theoretical flow of light field evolution and result reading. For scenarios involving univariate nth-degree differentiation and nth-fold integration, the temporal characteristics of this scheme differ significantly from traditional pure electronic numerical calculus schemes. This invention, relying on a unique optoelectronic collaborative computing architecture and a dual-temporal operation mechanism, reconstructs the temporal logic, completely resolving the shortcomings of traditional schemes such as explosive computing power, drastic latency increases, and error accumulation.

[0147] Traditional pure electronic numerical calculus schemes often employ finite difference, nested iteration, and grid fitting solution paradigms, resulting in time complexity reaching [missing information]. The computation time and computing power cost will increase exponentially or even exponentially with the increase of the degree of differentiation, the number of integrals, and the truncation precision. Under high-order heavy operation conditions, the problems of computing power explosion and latency surge are prominent.

[0148] This invention reconstructs the timing logic, completely solving this defect: the entire process of univariate nth-order differential is completed through fully parallel physical evolution of the laser light field, and the normal operation timing remains constant. It is not constrained by the degree of differentiation n, the truncation precision M, or the number of solution points; the core AC modal evolution operation of the univariate n-fold integral also maintains With constant timing, only a small number of integral remainders and boundary compensation terms are filled in by lightweight electronic closed-loop non-iterative computation, eliminating the problem of iterative timing expansion. The timing stability is significantly improved compared to traditional solutions, completely avoiding the industry pain points of high-order heavy computation, computing power explosion, and error accumulation. It is stable in operation and has excellent real-time performance.

[0149] 5.2 Extension and Verification of the Principles of Calculus for Multidimensional and Multivariable Functions This embodiment, based on the theoretical framework of univariate function solving, conducts dimensional expansion principle derivation and builds a D-dimensional multivariate function solving theoretical model, focusing entirely on the specific principle configuration and derivation process for multidimensional scenarios. Unlike the general multidimensional expansion mechanism described earlier, this section only elaborates on the specific selection, parameter configuration, isolation scheme, and practical logic of this principle derivation. Through concrete multidimensional operation principle derivation, it verifies the engineering feasibility mechanism of high-dimensional differentiated partial differential equations and multiple iterated integrals. This multidimensional expansion shares the same mathematical mechanism as the univariate solving system. Based on the unified mathematical rules, and according to the characteristics of D-dimensional multidimensional operations, a specific hardware system principle model adapted to high-dimensional coupled operations needs to be theoretically constructed. Through the theoretical design of matching multidimensional independent optical paths, multidimensional modal coupling structures, and a fully physical isolated hardware architecture, the hardware adaptation principle verification of D-dimensional multivariate calculus operations is achieved. This D-dimensional multivariate embodiment uses an electronic computing-assisted mode to complete the collaborative operation derivation throughout.

[0150] 5.2.1 Construction of Multidimensional Light Field Architecture and Derivation and Configuration of Specific Parameters This multi-dimensional principle derivation has a dedicated configuration logic: an independent benchmark period is matched for each dimension of the solution space. With respect to the corresponding fundamental frequency group (whose period is consistent with the corresponding dimension after adaptive normalization of the function to be solved), based on the Fourier series expansion formula of this embodiment, the d-th dimension reference fundamental frequency strictly satisfies the inherent relationship of Fourier orthogonal frequency domain: The fundamental frequency of higher harmonics in this dimension uniformly satisfies (k is the harmonic order index), the fundamental frequencies within a single dimension are pairwise orthogonal and have ordered gradients, and the fundamental frequency parameters of each dimension are independently adjustable, fully aligning with the multi-dimensional and single-dimensional independent fundamental frequency configuration mechanism.

[0151] In this embodiment, the multidimensional modeling process adopts a uniform truncation precision configuration throughout the entire domain. All solution dimensions adopt the same M-order symmetric truncation format, and the order index of each dimension takes a uniform value range of [-M,M]. The order is regular and uniform throughout the domain, and the modeling precision is consistent, avoiding the problems of mode number mismatch and basis asymmetry caused by multidimensional differential order matching.

[0152] After completing the theoretical design of the single-dimensional fundamental frequency initialization calibration, the principle of global coupling is deduced for all dimensions of fundamental frequency to generate multi-dimensional sub-fundamental frequency composite modes. After coupling is completed, the original single-dimensional fundamental frequency optical path is automatically shielded, and only the sub-fundamental frequency composite modes are retained to participate in the multi-dimensional operation principle deduction, strictly matching the core rules of multi-dimensional mode composition.

[0153] To address the theoretical pain point of crosstalk and aliasing caused by the superposition of multidimensional and multimodal modes, this embodiment uses a fixed and optimal physical isolation scheme with full coverage based on the principle deduction. At the same time, it is compatible with the adaptation logic of targeted physical isolation schemes. Both types of isolation architectures can be implemented as needed. It strictly aligns with the core protection mechanism of the dual-mode physical isolation of this invention, completely eliminating cross-dimensional and cross-modal crosstalk distortion from the mechanism level and ensuring the stability of high-dimensional operation theory.

[0154] Based on the aforementioned solidified principles, this multidimensional theoretical model can form three types of reusable high-dimensional solution capabilities: First, a multidimensional function-specific isomorphic modeling capability, which can accurately map and deduce multidimensional coupled modal optical fields for normalized multivariate functions; second, a multidimensional differentiated operator evolution capability, supporting independent configuration of differential degrees and integral multiplicity for any dimension, allowing individual application of calculus control to the target dimension while maintaining steady-state locking of modal parameters in other dimensions throughout, accurately matching the mathematical definition of multivariate partial calculus; and third, a multidimensional global analytical reading capability, which, after evolution, can simulate and read high-dimensional computation results at any spatial point, outputting the global analytical truth value. All capabilities are theoretical functions verified by this principle deduction and do not represent the completion of the physical prototype.

[0155] The core selection for this multidimensional principle derivation is fixed: Compared with targeted local isolation solutions, the full-domain, full-coverage isolation architecture, although slightly increasing the hardware protection cost, can fully adapt to complex computing conditions of high dimensions, multiple modes, and high orders, completely eliminate the problems of mode mixing and error amplification, and maximize the improvement of the theoretical accuracy and stability of multidimensional computing. Therefore, this isolation solution is used as the standard theoretical configuration throughout this principle derivation.

[0156] 5.2.2 Modeling and Derivation of Multidimensional Function Regularization and Multidimensional Light Field Principles This multidimensional principle derivation is based on the orthogonal expansion of Fourier series as its concrete theoretical foundation, establishing a fixed binding relationship between dimensional period, series expansion, and fundamental frequency parameters, ensuring that parameters are verifiable and configurable. For the multivariate function to be solved, this principle derivation employs a multidimensional-specific preprocessing theoretical process. Through multidimensional synchronous periodicity reduction, scale substitution, and interval truncation, complex multivariate functions are uniformly normalized into multidimensional standard periodic functions. For a very small number of complex coupled functions, an out-of-domain mapping fallback process is used to complete the adaptation and normalization principle derivation.

[0157] The derivation of this multidimensional modeling principle follows a fixed process: after function regularization, high-density discrete sampling is performed on the multidimensional global space to fully cover spatial information in all dimensions; relying on prior patented vector numerical optical computing, parallel optical computing is used to complete the calculation of all expansion series coefficients. This embodiment strictly distinguishes the boundary between optical field computation and electronically assisted computation, with clear division of labor and self-consistent mechanism, as detailed below:

[0158] 1. The calculations and mechanisms independently completed by the laser light field: A pure amplitude encoding method is used to map the absolute values ​​of the multidimensional coupling series coefficients to the amplitudes of each fundamental frequency optical path, thus completing the physical mapping modeling of the light field; relying on the optical detection and phase discrimination technology known in the field, the basic polarity determination of the optical path of the multidimensional fundamental frequency composite mode is completed. This polarity determination is a well-known backup technology in the industry and does not constitute the core innovation of this invention; the entire process relies on the multidimensional AC coupling mode to carry out the light field modeling and controlled evolution mechanism deduction, abandoning the pure DC mode optical path loading and optical path calculation, and only retaining the AC mode to participate in the light field physical calculation link.

[0159] 2. The electronic computing-aided system is responsible for the following operations: managing the external storage of positive and negative coefficients; combining the calculation results of each sub-fundamental band symbol output by the optical field, superimposing the positive and negative coefficients in the external storage to complete polarity correction, and summarizing and superimposing the full-domain calculation results by the electronic computing-aided system; undertaking all post-mathematical operations that the optical field cannot directly complete, such as the superposition calculation of full-domain modal results, the calculation of DC cumulative main terms, and the solution of multi-dimensional nested additional compensation terms.

[0160] Optical field physics calculations and electronically assisted computation work together, dividing tasks and collaborating to achieve accurate modeling and calculus verification of multidimensional multivariable functions. Multidimensional standard expansion: This multi-dimensional implementation only retains the AC-coupled mode for optical path calculation, while the pure DC term is only used for post-electronic supplementation and mathematical verification, and does not occupy optical field evolution resources.

[0161] The formula for solving the multidimensional coupling coefficient is:

[0162] In the formula, each parameter is a configurable concrete theoretical variable in this principle derivation, corresponding to the actual regularization period and order parameters of each dimension; there are no abstract theoretical parameters. In a multidimensional computational system with a single-dimensional truncation order of M and a total dimension of D, the order index of each dimension ranges from [-M, M], and the total number of coupled modes that can be generated across the entire domain is... It includes one set of pure DC zero-frequency modes and A group of AC-coupled modes. The number of effective optical paths for overall optical input and participation in optical field modeling and controlled evolution calculations in this invention is: , in the formula " The core meaning of "eliminating one group of DC zero-frequency modes that do not participate in optical path calculations and are only used for mathematical completeness verification and electronic supplementation, and retaining only all AC secondary fundamental frequency composite modes as effective input optical fields, thereby avoiding interference from invalid modes and simplifying the overhead of optical path calculation resources.

[0163] 5.2.3 Modulation of Multidimensional Optical Field Operators and Derivation of the Principle of Multivariable Calculus This multidimensional computation principle derivation strictly adheres to the solidified theoretical logic and core mechanism: based on the established multidimensional coupled optical field theoretical model, relying on the core mechanism of "single-dimensional directional evolution and multidimensional steady-state maintenance", it applies operator modulation to the target solution dimension in a directional manner for the user-preset differential order and integral multiplicity of each dimension, and locks the steady-state parameters of non-target dimensions throughout the process, which is in complete accordance with the mathematical definition of multivariable partial differential and multiple iterated integral. All operations are concrete and implementable parameter configuration and optical path control principle design, without general mechanism repetition.

[0164] This embodiment concretely implements a multidimensional differentiated order partial differential equation configuration, where the degree of differentiation in each dimension can be independently and freely set, adapting to non-homogeneous high-dimensional solution conditions. Any multivariate multiple derivative... The general formula for differentiation is as follows:

[0165] This implementation strictly adhered to practical rules. Pure DC terms did not participate in the optical field evolution, and pure DC modes were completely shielded. Global partial differential calculations were performed solely through multi-dimensional AC-coupled modes, ensuring a simple optical path and high computational efficiency. yes Dimensional variables of Second differential.

[0166] This embodiment presents a concrete construction site formula for multidimensional differentiated multivariate iterated integrals, adaptable to independent restatement configurations for each dimension, and supports any multivariate multivariate iterated integral. The general formula is as follows:

[0167] in yes Dimensional variables of Multiple integrals. This multidimensional integral principle derivation relies on a standardized optoelectronic division of labor theory model formed by electronic computing assistance: the series oscillation operation terms of high-dimensional coupled AC modes are completed by the parallel evolution theory of optical fields, with constant time sequence and no computational explosion; multi-dimensional nested boundary compensation terms, pure DC cumulative main terms, etc., are all uniformly and accurately completed by electronic computing assistance modules in a closed-loop, non-iterative manner. At the same time, polarity correction, global optical path superposition, and global scale correction are completed by electronic computing assistance, ensuring that the high-dimensional operation theory is rigorous and self-consistent and without cumulative errors. This electronically assisted collaborative division of labor model is the preferred solution for this multidimensional implementation.

[0168] This derivation of the multidimensional operator modulation principle follows specific, fixed rules, fully aligning with the core mechanism of modal independent modulation and one-to-one operator correspondence in this invention: each dimension's fundamental frequency corresponds one-to-one with its corresponding frequency operator, and the secondary fundamental frequency mode adaptively matches the corresponding operator to complete cooperative evolution based on its own coupling with the fundamental frequency component of the target dimension. Combined with the optimized global isolation architecture theoretical design, this provides dual protection for the independence and accuracy of high-dimensional operations, avoiding modal aliasing and computational distortion problems at the mechanistic level.

[0169] After the evolution principle is completed, the theoretical model can simulate and output a multidimensional global analytical light field, supporting result reading and deduction at any point and restoration of the true value. The running state timing is consistent with the unidimensional timing, the one-dimensional and multi-dimensional processes are unified, and the mechanism can be replicated, fully verifying the theoretical stability and universality of the high-dimensional extension of this invention.

[0170] 5.2.4 Reading and True Value Reconstruction of Multidimensional Light Field Evolution Results This section refines the closed-loop mechanism for reading the results of multidimensional multivariate calculus solutions. It builds upon the core processes of multidimensional light field modeling, operator modulation, and optoelectronic co-evolution described earlier. It strictly adheres to the unified result output paradigm, polarity determination rules, and truth value restoration technology system of this invention, adapts to the unique operating characteristics of multidimensional coupled modes, and completes the practical mechanism for reading the results in multidimensional solution scenarios. Ultimately, it realizes a replicable principle deduction system with a full-link closed loop of "modeling-evolution-computation-reading".

[0171] After the multidimensional light field completes the controlled evolution of the global coupled modes, it forms a continuous steady-state multidimensional analytical light field with the characteristics of continuous field value and phase distribution in the entire space. It can directly read the light field calculation parameters of any multidimensional spatial coordinate point without iteration or interpolation. The overall reading process distinguishes between two types of working conditions: partial derivative and iterative integral. At the same time, it is adapted to multidimensional exclusive polarity correction and true value restoration mechanism.

[0172] Multidimensional Partial Differential Equation (PDE) Result Reading: Multidimensional multivariate PDEs rely solely on AC-coupled sub-fundamental modes to complete the parallel evolution of the optical field, without pure DC leakage terms or boundary compensation terms, and without the need for electronic post-compensation terms. After the optical field evolution is completed, a layered interpretation process is adopted: "single optical path parameter reading - internal polarity determination - peripheral polarity calibration - global optical path result superposition". The field strength amplitude and phase shift after the evolution of each sub-fundamental optical path are collected, compared with the initial steady-state benchmark, and common knowledge is introduced to complete the basic polarity determination; the polarity is calibrated by combining the peripheral positive and negative algebraic labels preset by the optical path to obtain the accurate single-path calculation result; finally, the single-path results of all coupled modes are collected and superimposed globally to output the complete PDE calculation result of the target point, with no fitting, no iteration, and no accumulated error throughout the process.

[0173] Multidimensional iterated integration result reading: Multidimensional multi-iterated integration involves multidimensional nested boundary compensation terms that cannot be achieved by the optical field. A layered reading mode is adopted: "single optical path parameter reading - polarity determination and calibration - peripheral polarity calibration - global optical path result superposition - electronic closed-loop term supplementation". First, the amplitude and phase parameters after each fundamental frequency optical path evolution are read to complete basic polarity determination and precise calibration. Polarity is calibrated by combining the preset peripheral positive and negative algebraic markers of the optical path to obtain the independent calculation results for each path. All single-mode results are superimposed to obtain the total value of the original optical field oscillation calculation. Finally, the non-iterative closed-loop solution terms of the electronic system are superimposed to form a complete high-precision integration calculation result. There is no mesh iteration error or fitting distortion throughout the process, and the solution accuracy is precisely controllable by the truncation order M.

[0174] Multidimensional global truth value restoration output: Adopting three standardized output paradigms for univariate functions and matching a three-layer function adaptation system, it fully covers multidimensional operation scenarios: First, for natively matched multidimensional periodic coupled functions, the modeling is scale-free, and the results can be directly read after evolution without the need for correction or decoding; Second, for regularized multidimensional functions after periodic normalization and scale substitution, the truth value is restored by matching the calculus power rule through a globally unified pure algebraic scale correction formula, without additional decoding overhead; Third, for complex and special coupled functions, the modeling is completed by relying on the pre-domain external mapping, and the truth value is restored by decoding through the matching inverse mapping after operation, thus completing the bottom-line closure of the solution for high-dimensional special functions.

[0175] Core Features of Multidimensional Analytical Direct Reading: This invention's multidimensional solution possesses the core advantages of single-step global evolution and accurate direct reading at any point: Only one controlled evolution of the multidimensional coupled optical field is needed to generate a complete and continuous high-dimensional analytical optical field, eliminating the need for repeated iterative calculations at different points. Results for all target points within the solution domain can be read in batches and quickly. This completely solves the pain points of traditional multidimensional calculus, such as point-by-point traversal, computational redundancy, and interpolation distortion, significantly improving the accuracy of high-dimensional solutions and the efficiency of global computation.

[0176] 5.2.5 Analysis of the temporal complexity of multidimensional optoelectronic collaboration and comparison of computing power advantages To clarify the substantial computing power advantage of the photoelectric collaborative computing of this invention, define the computing power weights of the core optical field computing and electronic auxiliary terms, and avoid related technical doubts, this section focuses on two core computing scenarios: multidimensional multivariate partial differential and multiple iterated integral. It quantitatively decomposes the temporal complexity, analyzes the timing structure of photoelectric collaboration, and conducts a quantitative comparison with the traditional pure electronic numerical calculus scheme, comprehensively demonstrating the superior computing power and engineering innovation value of this invention.

[0177] This invention employs a heterogeneous computing power division model for solving multidimensional multivariate calculus problems, combining optical field main computation with electronic auxiliary term completion. The normalized total computation time is composed of the superposition of optical field parallel computation time and electronic auxiliary computation time. The general calculation formula is as follows: .in For the parallel physical evolution timeline of the laser light field, For electronically assisted compensation calculation timing, there are significant differences in the photoelectric division of labor and timing characteristics between the differential and integral operations, which need to be classified and analyzed separately.

[0178] Multidimensional Partial Differential Equation (PDE) Operation Time Sequence Decomposition (Pure Optical Field Constant Computing Power, No Electronic Overhead): Firstly, the optical field parallel operation time sequence: The entire multidimensional arbitrary-order PDE operation relies on multidimensional coupled sub-fundamental AC modes to complete a fully parallel physical evolution. Fixed-rule amplitude scaling and standardized phase shifting are only applied to the coupled mode components of the target solution dimension; non-target dimension modes remain steady-state throughout. This process is naturally parallel, unaffected by the number of dimensions D, the differential order n, or the truncation precision order M. It involves no iterative traversal, no layer-by-layer correction, and no incremental computing power overhead, resulting in a constant normalized time sequence complexity. Second, regarding the timing of electronically assisted computation, multidimensional partial differential equations have no pure DC leakage terms or nested boundary compensation terms. After the optical field evolution is completed, the global analytical differential result can be directly output without the need for post-compensation, accumulation, and correction operations by the electronic system, and the electronically assisted timing overhead approaches zero. Third, regarding the overall timing characteristics, the total timing of normalized multidimensional partial differential equation solutions is... It achieves constant-time operations that are completely independent of dimension, order, and precision.

[0179] Time series decomposition of multidimensional multivariate iterated integral computation (optical field as the main component + lightweight electronic component): First, the time series of parallel optical field computation, the core oscillatory mode evolution computation of multidimensional multivariate iterated integral, relies on all coupled sub-fundamental frequency AC modes to complete fully parallel non-iterative physical computation, consistent with the mechanism of time series differentiation. Increasing the number of dimensions, integral multiplicity, and truncation order will not cause time series expansion, and the time series complexity remains constant. This part carries the vast majority of the core modal evolution and global analytical field reconstruction workload for integral operations, and is the core computing power for high-precision integral solutions. Secondly, the electronically assisted computation timing: the electronic system only handles the closed-loop completion operations for multi-dimensional nested boundary compensation terms that cannot be physically realized in the optical field, and handles the pure DC cumulative main term as needed. There is no grid iteration, no point-by-point traversal, and no error iteration correction; it is only a single lightweight algebraic accumulation operation. Its timing is only weakly related to the truncation order and the multiplicity of the integral, satisfying low-order polynomial complexity. The computational cost is extremely high, and there is no cumulative time redundancy. Thirdly, regarding the overall time characteristics, the total time of the multidimensional, multivariate, iterative integral solution is... By superimposing low-order polynomial timing, the overall growth is only controllable low-order polynomial, completely avoiding the risk of computing power explosion.

[0180] Comparison of time-series computational power with traditional pure electronic solution schemes: First, a comparison of computational power for multidimensional partial differential equations. Traditional pure electronic multidimensional partial differential equations generally adopt finite difference and finite element discrete iterative paradigms. For D-dimensional multivariate, M-order high-precision, and n-order partial differential equations, it is necessary to traverse the entire discrete mesh point by point, iterate step by step, and correct errors layer by layer. The time-series complexity increases exponentially with dimension and accuracy, which is insufficient to meet the requirements of traditional pure electronic solution schemes. (N is the number of discrete grid points in the entire domain; in high-precision applications, N can be in the tens of thousands.) In high-precision applications, latency increases dramatically, computational costs soar, and inherent truncation and cumulative errors exist, making it impossible to simultaneously achieve both accuracy and timeliness. Furthermore, traditional numerical algorithms only support point-by-point iterative calculations. To obtain results from multiple locations across the entire domain, the entire iterative calculation process must be repeated for each target point, resulting in repeated computational overhead, a cumbersome process, and extremely low efficiency. This invention maintains [a certain level of precision / efficiency throughout the entire process]. With constant timing, regardless of upgrades in dimension, order, or precision, the computational cost and latency remain constant, with no iterative accumulation error. It can generate a complete and continuous analytical light field by relying on the parallel evolution of the entire light field in a single operation, and complete the calculation and solution of all points in the entire field in one go. Subsequently, the true value of any target point can be directly read without iteration or repetitive calculation, completely overturning the inherent defects of traditional electronic differential equations, such as point-by-point processing, repeated iteration, and computational redundancy.

[0181] Secondly, comparing the computational power of multidimensional iterative integration, traditional pure electronic multidimensional iterative integration uses a grid nested iteration and numerical approximation fitting scheme, resulting in an exponential increase in time complexity. In high-dimensional conditions, the amount of mesh traversal increases exponentially, leading to an explosive increase in computational overhead and solution latency. Furthermore, the continuous accumulation of errors across multiple iterations makes achieving high-precision and high-stability solutions extremely difficult. Traditional solutions can only perform single-point successive calculations; multi-point and global solutions require repeated nested integration iterations, resulting in significant redundant computational consumption and extremely low global solution efficiency. This invention uses constant-time optical field computation as its core and lightweight electronic complementation as an auxiliary method. The temporal increase is only slightly controllable, while the optical field output analytical solution and electronic complementation are performed using non-iterative closed-form precise computations, with no fitting distortion or error accumulation throughout the process. The entire solution domain integration computation can be completed with a single global optical field evolution, covering all coordinate points in the entire domain at once. There is no need for separate iterations for each target point, and the global result reading incurs no redundant computational overhead, balancing ultra-high solution accuracy with real-time computational efficiency.

[0182] This invention employs a differentiated optoelectronic heterogeneous division of labor model, effectively addressing industry pain points such as computational power explosion, cumbersome iterations, and error accumulation inherent in traditional pure electronic calculus algorithms. The division of computational power is clear, and the technological gains are significant: multidimensional partial differential equations rely on a constant time sequence of pure optical fields to complete fully parallel computations without the intervention of electronic computing power; multidimensional iterative integrals primarily utilize constant-power optical field parallel evolution, supplemented by lightweight electronic closed-loop term completion, with the optical field serving as the core computational power carrier, demonstrating outstanding technological innovation. This invention uniquely establishes a solution paradigm of single-cycle global optical field evolution, one-time global solution, and precise direct reading from any point, abandoning the cumbersome process of single-point iteration and multi-point repetitive computation in traditional algorithms, significantly reducing computational power redundancy and computational latency. The overall architecture achieves a computing power upgrade with constant differential timing and controllable weak timing growth of integrals. In high-dimensional, high-order, high-precision, and multi-point engineering solution scenarios, it has the core advantages of stable latency, low computing power overhead, no cumulative error, and high efficiency of global solution. It has overcome the technical challenges of poor accuracy, weak real-time performance, and low efficiency of multi-point solution in high-dimensional calculus, and has extremely high engineering application value and technological innovation.

[0183] 5.3 Evidence of the Principles, Effects, and Innovation of the Examples This embodiment uses the most versatile Fourier orthogonal series as a concrete verification vehicle to fully deduce the principle of solving calculus problems for univariate and finite D-dimensional multivariate functions under all working conditions. It covers the entire process of optical path initialization, function regularization, optical field modeling, operator modulation, optoelectronic collaborative computation, result restoration, and computing power analysis, constructing a standardized verification system with configurable parameters, replicable processes, and traceable mechanisms. It should be clarified that this Fourier series embodiment is merely an exemplary application case of this patent, used to demonstrate the theoretical feasibility, mechanism stability, and working condition adaptability of the core architecture of this invention. It does not constitute any limitation on the scope of protection, implementation form, or technical adaptation system of this invention. The core technical mechanism of this invention can be seamlessly transferred and adapted to all series systems with orthogonality and completeness, exhibiting extremely wide applicability and engineering application scenarios.

[0184] This concrete and quantitative deduction of principles and comparison of computing power fully demonstrates the seven core technical features and engineering application value of this invention. It fully corresponds to all the core innovative mechanisms mentioned above, including the seven underlying functional units, three-layer engineering function adaptation, single-dimensional control and multi-dimensional steady-state maintenance, independent modal evolution, dual-mode physical isolation, dual-time-series efficient operation, and optoelectronic collaborative computing power division. It systematically solves the inherent technical pain points of traditional pure electronic numerical calculus algorithms, such as computing power explosion, error accumulation, poor real-time performance, weak adaptability, and insufficient scalability. The seven core standardized features that distinguish this invention from traditional numerical solution techniques have been fully implemented and verified as follows: First, the function adaptation scenarios are broad. Relying on a three-layer differentiated function adaptation system of precise matching adaptation, conventional preprocessing regularization, and out-of-domain mapping as a fallback, it comprehensively covers various engineering functions such as periodic, aperiodic, time-varying, and multi-coupling functions. It is compatible with multiple series expansion formats with orthogonality and completeness, and can flexibly balance the solution accuracy and computational efficiency through finite-order symmetric truncation, adapting to all types of conventional engineering solution conditions. Second, the order solution adaptation is flexible, relying on modally independent and controllable calculus operator tuning. The mechanism eliminates the need for iterative fitting and reconstruction of computational logic, allowing for on-demand solutions to arbitrarily finite-degree high-order differentials and arbitrarily finite-multiple-number integrals. Solution accuracy is autonomously controllable by the truncation order M, covering various engineering needs from lightweight rapid calculations to conventional accuracy solutions and ultra-precise high-order solutions. Thirdly, it boasts strong variable dimensional extensibility. Based on the core original mechanism of independent single-dimensional control and multi-dimensional steady-state maintenance, it achieves lossless and homogeneous expansion of multivariate functions from univariate to any finite D-dimensional multivariate functions. The operator control logic for each dimension is independently controllable, strictly conforming to the mathematical definitions of multivariate partial differentials and multiple iterated integrals, and is dimensionless. Fourth, it unifies the calculus operation system, breaking the industry drawbacks of the traditional fragmented differential and integral operation architectures. Both types of operations share the same set of laser light field modeling, evolution, and solution paradigms. The underlying mathematical rules and hardware operation system are completely homogeneous, realizing integrated closed-loop solution of calculus; Fifth, it allows free switching of operation operators. Based on the unified light field evolution architecture, it can flexibly switch between standardized differential and integral operators without changing the hardware architecture and operation process. It is bidirectionally adaptable to various calculus solution conditions and has a very wide range of engineering applications; Sixth, it unifies the high- and low-order heavy operation paradigms. The system employs a completely consistent optical field modulation and evolution process for arbitrary-order differentials and arbitrary-multiple integrals. High-order and high-multiple operations do not require system reconstruction or process modification, resulting in zero-cost expansion and extremely strong versatility. Seventh, it achieves ultimate global analytical direct-reading accuracy, overturning the traditional solution paradigm of grid iteration and discrete fitting. It generates a continuous steady-state analytical optical field through a single controlled evolution of the global optical field, enabling direct reading of accurate true values ​​at any point in the solution domain without iteration, interpolation, or fitting. This completely eliminates the defects of traditional algorithms such as truncation error, interpolation distortion, and iterative accumulation error, achieving analytical-level high-precision solutions.Overall, this invention features standardized principles and processes, flexibly configurable core parameters, and a universal and reusable hardware architecture. It achieves constant computing power output through a dual-time-sequence operation mechanism and ensures stable operation across the entire domain through a dual-mode physical isolation architecture. Its engineering adaptability, scenario scalability, computational accuracy, and real-time computing power efficiency are significantly better than traditional numerical solution schemes. It can be widely applied in high-precision engineering scenarios such as industrial simulation, multi-field coupling analysis, high-precision dynamics solution, and high-end numerical inference.

[0185] The scope of protection of this invention is not limited to Fourier series verification schemes. Any system based on the core original architecture of this invention that reuses the invention's original seven-layer full-dimensional full-coverage calculus solution closed-loop system, three-layer differentiated engineering function adaptation and regularization system, standardized calculus operator global unified regulation, single-mode independent controlled modulation evolution, single-dimensional directional evolution multi-dimensional steady-state maintenance core extension mechanism, dual-mode standardized physical isolation anti-crosstalk architecture, initialization modeling and normalized operation dual-time-series differentiated operation mechanism, optoelectronic collaborative heterogeneous computing power division of labor computing architecture, global analytical light field single evolution accurate direct reading at any point, multi-scene layered truth value restoration output system, and seven-layered underlying functional unit construction system (standardized orthogonal series adaptation system and fundamental frequency mode set, global amplitude and phase physical benchmark unified calibration) is also protected. All core proprietary technologies and architectures, including the quasi-system, independent optical path modulation channel configuration system, dedicated calculus operator system, standardized optical field coding mapping unit, high-precision full-domain optical field reading unit, and optical field anti-crosstalk physical isolation unit, are adapted to any orthogonal series system, any finite D-dimensional solution dimension, any finite degree higher-order differential, any finite multiplicity multiple integral, and single- or multi-dimensional full-condition coverage high-precision, error-free, constant low-latency laser field controlled evolution calculus solution, through equivalent replacement of orthogonal series system, engineering implementation process adaptation and optimization, migration and reuse of underlying mathematical mechanisms, equivalent modification of hardware architecture, reconstruction of operation timing logic, and equivalent replacement of operator regulations. All of these technologies fall within the protection scope of this invention. Attached Figure Description

[0186] Figure 1 This is a flowchart illustrating the overall implementation process of the present invention for calculus calculation based on controlled evolution of a laser field; Figure 2 This is a schematic diagram illustrating the principle of independent mode modulation and calculus-controlled evolution of one-dimensional orthogonal fundamental frequency optical field in this invention. Figure 3 This is a schematic diagram illustrating the multidimensional architecture, coupled modes, and multidimensional calculus-controlled evolution principle of this invention based on a one-dimensional system extension. Figure 4 This is a schematic diagram comparing the computational power characteristics of the dual-time-sequence operation mechanism of this invention with those of the traditional numerical iterative algorithm.

[0187] Figure 1This invention demonstrates a standardized four-step closed-loop solution process: S1 Constructing a laser field state evolution space adapted for calculus calculations; S2 Isomorphically representing the laser physical field of the calculus function to be solved; S3 Performing controlled evolution of the laser physical field, equivalent to calculus numerical calculations; and S4 Reading the calculus calculation results from the target state of the laser physical field. This closed-loop process is adaptable to calculus solutions for one-dimensional, multi-dimensional, multiplicative, and multi-type engineering functions, and serves as the top-level implementation architecture for the overall technical solution of this invention.

[0188] Figure 2 This paper demonstrates the arrangement characteristics and calculus evolution mechanism of orthogonal fundamental frequency laser fields in a one-dimensional scene. A single-dimensional configuration includes one zero-frequency substrate and two M symmetrical harmonic substrates, with orthogonal modes free from crosstalk. By applying differential power amplification, integral power attenuation, and standard phase shift to the independent fundamental frequency optical fields, one-dimensional high-precision calculus solutions are achieved, reflecting the core principles of independent single-dimensional mode evolution and unified operator rules in this invention.

[0189] Figure 3 This diagram illustrates the multidimensional architecture, coupled modes, and controlled evolution principle of multidimensional calculus based on the one-dimensional system extension of this invention. Using three-dimensional space as a concrete example, it demonstrates the hardware architecture, mode generation rules, and complete computational flow for extending a one-dimensional system to a finite D-dimensional multi-element system. The multidimensional extension process of this invention adheres entirely to the one-dimensional homogeneous mathematical rules, maintaining orthogonal decoupling of the fundamental frequencies in each dimension. The secondary fundamental frequency optical paths generated by coupling are independent and free from crosstalk and dimensional extension distortion; the global coupled modes follow... Power-order complete expansion ensures accurate and complete multidimensional modeling. The multidimensional computation system inherits the core mechanisms of independent evolution of one-dimensional modes and precise operator control. It can apply calculus operator evolution to the target dimension in a directional manner, while maintaining the steady state of the light field in other dimensions. It accurately matches the mathematical definitions of multivariate partial differential equations and multiple iterated integrals, fundamentally solving industry pain points such as computational power explosion, dimensional coupling interference, and low solution accuracy in traditional high-dimensional calculus solutions.

[0190] Figure 4 The computational time-series characteristics of this invention are compared with those of traditional numerical iterative algorithms. This invention employs a dual-time-series architecture of initialization modeling and constant normalization computation. The time-series characteristics shown in the figure only apply to the main optical field computation, and its normalization computation time-series overhead remains constant. The computing power and latency do not increase with the increase of solution dimensionality and precision; the timing of the auxiliary electronic compensation operation is extremely low, far superior to the computing power overhead of traditional all-electronic numerical computing. The computing power of traditional numerical iterative algorithms increases exponentially with the increase of dimensionality and precision, which intuitively highlights the computing power and real-time advantages of this invention in high-dimensional, high-precision engineering solution scenarios.

Claims

1. A method for calculus calculation using controlled evolution of a laser field, characterized in that, The solution for multivariate partial differential equations and multiple iterated integrals of finite-dimensional multivariable functions generally includes the following steps: S1. Constructing a laser field state evolution space adapted for calculus calculations: Each solution dimension is independently configured with its own dedicated fundamental frequency group. Within a single dimension, the fundamental frequencies are pairwise orthogonal and their frequency gradients are ordered. The frequency distribution pattern of each dimension's fundamental frequencies can be bidirectionally adaptively matched with the inherent periodic characteristics and composite modal series expansion of the function to be solved in the corresponding dimension. Amplitude and phase reference standardization initialization calibration is uniformly performed on all independent fundamental frequencies across all dimensions, and the global modal initialization is achieved through amplitude normalization. The system establishes an initial amplitude reference and unifies the global phase zero-position reference by returning the phase to zero, eliminating initial deviations and phase shift errors in the optical path. It couples the independent orthogonal fundamental frequencies configured in each dimension globally to generate a multi-dimensional sub-fundamental frequency composite mode optical path with a zero-phase reference. Each sub-fundamental frequency optical path is independent and complete, and each sub-fundamental frequency optical path is pre-bound with peripheral positive and negative algebraic identifiers. It constructs three core capabilities: function isomorphism expression, independent modulation and controlled evolution of calculus operators, and accurate reading of optical field results. It also configures two types of physical isolation protection mechanisms: targeted isolation of high-risk modes and full-coverage isolation of all modes. S2. Isomorphic representation of the laser physical field of the differential function to be solved: Under the constraints of preset engineering accuracy and solution order, relying on a three-layer differentiated adaptation system of precise matching adaptation, conventional preprocessing regularization, and out-of-domain mapping fallback, preprocessing is performed on various engineering functions that can be solved by finite-order truncation or can be solved by finite-order truncation after regularization. They are uniformly transformed into a standard series form based on multidimensional coupled sub-fundamental frequency composite modes. The multidimensional series coupling coefficients are obtained to complete the standardized modeling. Using a globally unified amplitude encoding rule, the amplitude of the sub-fundamental frequency optical path is modulated to the absolute value of the corresponding series coefficients. The positive and negative signs of the corresponding series coefficients are independent algebraic identifiers on the periphery and do not participate in optical path encoding and phase modulation, thus realizing the standardized isomorphic representation of the engineering function in the laser light field. S3. Controlled Evolution of Laser Physics Field Equivalent to Numerical Calculus: Dedicated frequency-point calculus operators are matched for each dimension of the fundamental frequency. The multidimensional coupled sub-fundamental frequency mode is the only effective evolution unit, strictly following the core mechanism of "single-dimensional directional evolution and multi-dimensional steady-state maintenance". By adapting the standardized operators for differentiation and integration, independent and differentiated amplitude and phase control is implemented on the dimensional components of each sub-fundamental frequency optical path coupling, completing the precise controlled evolution of the global mode. This is equivalent to realizing precise numerical calculation of multivariate multiple partial differentials and multiple accumulated integrals. The entire process maintains independent mode control, no crosstalk, and no dimensional coupling interference, strictly conforming to the mathematical definition and operation rules of multivariate calculus. S4. Laser physical field target state reading calculus calculation results: After the multidimensional controlled evolution of the optical field is completed, the field values ​​of the target coordinate points of each fundamental frequency optical path are read and the polarity change inside the optical path is determined. Combined with the positive and negative labels of the outer periphery bound to the optical path, the single optical path calculation value is obtained. The pure optical field calculation value is obtained by superimposing the results of the electronic closed-loop auxiliary supplementary calculation under the integral condition to complete the output of the full-domain calculation result. Corresponding to the three-layer function adaptation system, three types of truth value restoration methods are configured: direct reading output, pure algebraic multiplier correction output, and on-demand inverse mapping decoding restoration output.This method constructs a seven-layer global solution system, which includes function adaptation, order solving, variable dimension, operation system, operator switching, multiple operation paradigms, and global analysis direct reading, to adapt to all working scenarios of multidimensional multivariate multiple partial differential equations and multiple iterated integrals.

2. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, The S1 stage is further defined as follows: S1 is the global, bottom-level initialization stage for laser field solving. It establishes and adapts seven underlying functional units to multi-dimensional computation scenarios, including global amplitude and phase reference calibration, standardized orthogonal series and fundamental frequency mode system, independent optical path modulation channel, dedicated calculus operator system, optical field encoding and mapping unit, high-precision optical field reading unit, and dual-mode optical field anti-crosstalk isolation unit, forming a complete multi-dimensional laser field solving support system. This stage involves independently configuring dedicated fundamental frequency groups with orthogonal order and adaptive matching function characteristics for each dimension. After completing amplitude normalization and phase zeroing calibration of the single-dimensional fundamental frequency, a multi-dimensional secondary fundamental frequency composite mode optical path is generated through global coupling. After coupling, the single-dimensional fundamental frequency exits the computation link, and only the secondary fundamental frequency composite mode serves as the effective carrier for multi-dimensional computation. It can also directly construct the secondary fundamental frequency composite mode through standardized mathematical modeling methods without the traditional step-by-step coupling process of the single-dimensional fundamental frequency, adapting to diverse laser field modeling and computation scenarios. This phase establishes three core capabilities: isomorphic light field representation, modal independent and controlled evolution, and precise global point reading. A dual-time-series operational framework is built, with the normalized computation timing remaining constant. It features a dual-mode physical isolation architecture to avoid multimodal aliasing and optical field crosstalk, ensuring stable and accurate computation.

3. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, S2 uses multidimensional coupled sub-fundamental frequency composite modes as its exclusive modeling basis, strictly following the five-step standardized core process of "function regularization - discrete sampling - parallel parameter calculation - amplitude and phase encoding - mode superposition and shaping". It relies on three layers of differentiated adaptation logic to complete the standardized optical field isomorphic representation of multidimensional engineering functions; it can selectively include pure DC zero frequency into the optical field evolution; it calculates the series expansion coefficients of all multidimensional coupled modes, modulates the absolute value of the coupling coefficients into the corresponding sub-fundamental frequency optical path amplitude, and the positive and negative signs of the coefficients are independent algebraic attributes of the periphery, realizing the accurate physical isomorphic representation of multidimensional engineering functions.

4. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, S3 adheres to the core rules of "one-to-one correspondence between fundamental frequency and frequency point operator, directional evolution of the target dimension, and steady-state maintenance of non-target dimensions." It sets n to a positive integer greater than or equal to 1, enabling it to perform calculus operations of different orders and multiplicity through operator iterations or overall synthesis. The nth derivative relies on the nth power amplification of the optical path amplitude at the corresponding modal fundamental frequency point, matching an n-fold positive... Standardized phase shift allows for parallel evolution calculations across the entire differential process relying solely on the optical field's AC modes, eliminating electronic compensation overhead. Integral calculations employ differentiated operating condition regulations: during coupling, the secondary fundamental frequency mode, lacking the target dimension's zero-frequency component, is used. The nth integration leverages the nth power reciprocal of the corresponding mode's fundamental frequency to achieve inversely proportional attenuation of the optical path amplitude and match n times the reverse frequency. Normalized phase shift, coupling contains the target dimension zero-frequency secondary fundamental mode, according to A fixed multiple is used to complete pure amplitude modulation of the optical path without performing phase shift control. The multi-dimensional nested boundary compensation remainders corresponding to the integration operation are accurately completed by electronic closed-loop non-iterative computation, with no iteration error or fitting distortion throughout the process, ensuring controllable accuracy of the integration operation across the entire domain. It can selectively exclude pure DC terms from the optical field evolution and complete them using electronic computation.

5. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, S4 employs a dual-layer polarity determination logic. First, it determines the basic polarity of a single-path secondary fundamental frequency optical path. Then, it superimposes the external independent positive and negative algebraic identifiers to obtain the single-path calculation result. The total value of the optical field calculation is obtained by summing the results across the entire domain. It is configured with three types of truth value restoration paradigms corresponding to the three-layer function adaptation system: conventional regularization function adaptation with adaptive pure algebraic scale correction, and special function adaptation with inverse mapping decoding restoration. The differential is directly restored to the output, and the integral is superimposed with the electronic complement result before performing the truth value restoration. Multidimensional coupled mode polarity ambiguity is corrected using known optical techniques without intervening in the core optical field computation link.

6. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, The solution of univariate single-dimensional calculus has unique core characteristics: it does not require the construction of multi-dimensional coupled sub-fundamental modes, but directly adopts orthogonal series expansion and completes the calculation based on a single-dimensional independent orthogonal fundamental mode; the univariate function operation scenario can adopt an amplitude and phase joint encoding mechanism, which can encode the absolute value and positive and negative polarity of the series coefficients into the single-dimensional fundamental optical field, and complete the mapping of the full-domain information of the coefficients through the joint modulation of optical path amplitude and phase, thus completing the mapping of the built-in parameters of the optical path; The single-dimensional optical path can autonomously and unambiguously complete polarity determination based on its own steady-state field shape and phase extremum characteristics without the need for auxiliary correction; the calculus operator directly acts on the single-dimensional fundamental frequency to complete the equivalent evolution operation. After the optical field evolution is completed based on the amplitude-phase joint encoding mechanism, the complete calculus operation result with positive and negative signs can be directly read without additional post-coefficient multiplication or post-parameter compensation operations; there is no cross-dimensional modal crosstalk in the one-dimensional scene, the fundamental frequency modes of each order are orthogonal and independent, and the mathematical mechanism and core rules of univariate and multidimensional operations are completely homogeneous, with only the encoding implementation form being different.

7. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, The four-step optical field isomorphic evolution architecture features modular decoupling, specifically including: modal unit decoupling, dimensional control logic decoupling, computational branch decoupling, dimensional scene decoupling, function adaptation system decoupling, core parameter decoupling, and auxiliary architecture decoupling. Each decoupling unit can be separately decomposed, independently implemented, and layered for defense. Modal unit decoupling refers to the independence and lack of crosstalk between fundamental and sub-fundamental frequency optical field modes, with a single mode as the smallest independent computational unit. Dimensional control logic decoupling refers to the independence of fundamental frequency configuration, operator control, and steady-state maintenance logic for each solution dimension, supporting independent evolution of a single dimension and steady-state operation in other dimensions. Computational branch decoupling refers to the separation of differential computation optical path branches from integral operations. The optical path branches are completely decoupled, with the two types of computation links being independently controlled and unaffected by each other; dimensional scene decoupling refers to the layered decoupling of the architecture between unidimensional and multidimensional computation scenarios, where the scene mechanisms are of the same origin but the operating logic can be independently adapted; function adaptation system decoupling refers to the three types of layered function adaptation mechanisms being independent of each other and each performing its own function, adapting to the regular modeling of engineering functions with different characteristics; core parameter decoupling refers to the fact that the core parameters such as truncation order, derivative degree, integral multiplicity, and number of dimensions can be independently configured and are not bound to each other; auxiliary architecture decoupling refers to the functional decoupling of the physical isolation, amplitude and phase calibration, and polarity correction auxiliary systems from the core optical field computation architecture, independently ensuring computational stability and accuracy.

8. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, Two types of standardized physical isolation protection architectures that can be independently deployed are configured. The first type is targeted physical isolation, which implements fixed-point physical isolation for high-order single-dimensional fundamental frequency components and multi-coupling secondary fundamental frequency components that have the risk of crosstalk and aliasing. The second type is full-domain, full-coverage physical isolation, which implements unified isolation protection for all single-dimensional fundamental frequency components and multi-coupling secondary fundamental frequency composite modes.

9. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, Step S2 relies on the three-layer differentiated engineering function adaptation system of this invention to complete the hierarchical preprocessing and normalization of single-variable and multi-dimensional variable engineering functions and the standardized optical field isomorphic modeling, realizing the accurate isomorphic mapping of various compliant engineering functions and multi-dimensional composite modal optical fields and the physical expression of amplitude encoding. Step S2 includes several independent and decoupled core procedures, specifically including unified calibration of the global fundamental frequency amplitude and phase reference, preprocessing and normalization of the target engineering function shape, equivalent period reconstruction adaptation, matching and configuration of single-dimensional / multi-dimensional orthogonal wave function groups, accurate solution of multi-dimensional coupling series coefficients, secondary fundamental frequency mode amplitude encoding mapping, and fixed configuration of modal operation parameters. Each core procedure is functionally independent and unbound from each other, without fixed time sequence constraints, and can be flexibly arranged according to engineering implementation requirements, executed step by step or processed in parallel, adapting to differentiated hardware deployment and modeling operation conditions.

10. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, It is fully compatible with two types of complete basis expansion methods: one-dimensional orthogonal fundamental frequency wave function groups and multi-dimensional composite modal secondary fundamental frequency wave function groups. It follows a unified modeling paradigm of "single-dimensional orthogonal basis first, multi-dimensional coupled basis expansion". One-dimensional scenarios are adapted to orthogonal series expansion, and multi-dimensional scenarios are adapted to composite modal series expansion. The frequency point arrangement rules of each dimension are adaptively matched with the periodic characteristics and series expansion forms of the function to be solved in the corresponding dimension. The legal adaptation boundary is defined: within the engineering accuracy range, single-variable one-dimensional and finite multivariate multidimensional engineering functions that can be expanded by finite-order truncated orthogonal series expansion, normalized finite-order truncated orthogonal series expansion, or normalized finite-order truncated orthogonal series expansion after external mapping are all included in the protection range. One-dimensional and multi-dimensional operation scenarios that cannot achieve finite-order orthogonal series expansion or do not meet the engineering truncation convergence conditions are excluded.

11. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, This invention excludes the protection scope of known technologies such as one-dimensional / multi-dimensional orthogonal wavefunction expansion, conventional function preprocessing, and general-purpose optical hardware; the core protections include a proprietary four-step closed-loop calculus solution architecture, a seven-layer full-dimensional modular decoupling architecture, seven underlying functional units, a one-dimensional orthogonal fundamental frequency and multi-dimensional coupled secondary fundamental frequency dual-layer wavefunction group construction mechanism, a three-layer hierarchical function adaptation and regularization system, a single-mode independent evolution mechanism, a single-dimensional directional evolution multi-dimensional steady-state maintenance mechanism, a dual-mode physical isolation architecture, a dual-temporal differentiated operation mechanism, and a multi-scenario hierarchical truth output system; equivalent substitutions, temporal reconstructions, and equivalent architectural modifications based on the aforementioned core architecture and core mechanisms all fall within the protection scope of this invention.

12. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, The system is equipped with two independently switchable engineering operation modes: an independent operation mode and an optoelectronic collaborative operation mode. In the independent operation mode, the differential is calculated entirely by the pure optical field, the integral optical field completes the AC mode calculation, and the electronic closed-loop calculation completes the remainder terms. The pure DC term can be implemented by either the optical field evolution or the electronic calculation. In the optoelectronic collaborative operation mode, the system adopts an optoelectronic heterogeneous computing power collaborative offloading architecture. The complex multimodal core evolution calculation is carried out in parallel by the optical field, while the lightweight simple calculation, boundary compensation, and scale correction tasks are collaboratively shared by the electronic system. The two modes rely on a dual-time sequence mechanism and a differentiated optoelectronic division of labor system to adapt to different computing power load conditions, achieving high-precision equivalent solution across the entire domain.

13. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, This invention possesses seven core technical features: Dimensional global adaptability, enabling computation from unidimensional to any finite D-dimensional dimension; unidimensional and multidimensional computation mechanisms share the same origin and paradigm; Flexible and adjustable precision, relying on the truncation order M to adjust precision as needed; Adaptive matching of frequency point arrangement and function characteristics across dimensions; Constant timing, with the core evolution maintaining a normalized timing sequence. No computing power explosion, no error accumulation; It supports global calculus calculations, with a single architecture compatible with differentials of any order and integrals of any multiplicity; it features a unified operational paradigm, eliminating the need for reconstruction of high- and low-order or high- and low-multiplicity operational architectures; it provides global analytical direct reading, generating analytical light fields from a single light field evolution, allowing for precise value acquisition at any point without discrete iteration errors; it is stable and interference-resistant, featuring a dual-mode standardized physical isolation architecture that completely avoids mode aliasing and light field crosstalk caused by multi-mode superposition, ensuring high-precision and high-stability computation under all working conditions.

14. The method for calculus calculation using controlled evolution of a laser field according to claim 1, characterized in that, This invention does not limit whether the series coefficients are pre-encoded by optical path or the specific implementation of pre-encoding, and is compatible with all feasible optical path mapping and operation implementation methods. In particular, the univariate function operation scenario can directly input the amplitude and polarity of the series coefficients, and directly read the signed operation result after the optical field evolution, which is suitable for the lightweight and simplest implementation mode.

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