A task processing method, apparatus, device and product

Abstract

The application provides a task processing method, device, equipment and product, the method comprises the following steps: obtaining a modeling parameter required when a target task is modeled by a Sturm-Liouville equation, the target task comprises at least one of an elastic modal analysis task, a heat conduction analysis task, an electromagnetic wave propagation characteristic analysis task and a particle state analysis task; solving a characteristic parameter of the Sturm-Liouville equation according to the modeling parameter and a physical law corresponding to the target task, obtaining a task processing result, the characteristic parameter characterizes a solving target corresponding to the target task, and the solving target satisfies the physical law. The application can improve the accuracy of the processing result of the physical modeling task.

Description

Technical Field

[0001] The present application relates to the field of artificial intelligence, and in particular to a task processing method, apparatus, device and product. Background Art

[0002] In physical system modeling tasks such as engineering structural mechanics and vibration analysis, heat conduction modeling, and electromagnetic field simulation, the Sturm-Liouville problem, a typical eigenvalue problem for second-order linear differential equations, is widely used to describe the intrinsic characteristics of a system, such as natural frequency, modal shape, or energy distribution. Solving this type of problem enables modeling and prediction of the dynamic behavior of physical systems, providing a theoretical basis for equipment condition monitoring, fault diagnosis, and performance optimization.

[0003] With the development of artificial intelligence (AI) technology, machine learning methods are increasingly being used to solve related modeling tasks, aiming to improve computational efficiency and adapt to complex boundary conditions. However, in practical applications, machine learning models still suffer from issues such as insufficient generalization under conditions outside the training data distribution, difficulty in effectively incorporating fundamental laws of physical systems (such as conservation laws or boundary constraints) as prior knowledge, and instability during the numerical solution process. These issues lead to low-precision results for physical modeling tasks. Summary of the Invention

[0004] Based on the above-mentioned technical status, the present application provides a task processing method, device, equipment and product, which can enhance the physical consistency of task processing results and improve the accuracy and reliability of task processing results.

[0005] In order to achieve the above technical objectives, this application specifically proposes the following technical solutions:

[0006] According to a first aspect of an embodiment of the present application, a task processing method is provided, comprising: obtaining modeling parameters required for modeling a target task through the Sturm-Liouville equation, wherein the target task includes at least one of an elastic modal analysis task, a heat conduction analysis task, an electromagnetic wave propagation characteristic analysis task, and a particle state analysis task; solving characteristic parameters of the Sturm-Liouville equation according to the modeling parameters and the physical laws corresponding to the target task to obtain a task processing result, wherein the characteristic parameters represent a solution target corresponding to the target task, and the solution target satisfies the physical laws.

[0007] According to a second aspect of an embodiment of the present application, a task processing device is provided, including: a task processing unit, configured to implement the task processing method as described in the first aspect.

[0008] According to a third aspect of an embodiment of the present application, an electronic device is provided, comprising a memory and a processor; the memory is connected to the processor and is used to store programs; the processor is used to implement the task processing method described in the first aspect by running the program in the memory.

[0009] According to a fourth aspect of an embodiment of the present application, a computer program product is provided, comprising computer program instructions, which, when executed by a processor, cause the processor to execute: the task processing method as described in the first aspect.

[0010] The embodiment of the present application provides a task processing method, device, equipment and product. The method first obtains the modeling parameters required for modeling the target task through the Sturm-Liouville equation. The target task includes at least one of the elastic modal analysis task, the heat conduction analysis task, the electromagnetic wave propagation characteristic analysis task and the particle state analysis task. Afterwards, the characteristic parameters of the Sturm-Liouville equation are solved according to the obtained modeling parameters and the physical laws followed by the target task to obtain the task processing result; the characteristic parameters characterize the solution target corresponding to the target task, and the solution target is consistent with the corresponding physical law. In this technical solution, the target task is modeled through the Sturm-Liouville equation, and the target task is processed by solving the Sturm-Liouville equation. Since the physical laws related to the target task are introduced in the solution process, the task processing result satisfies the corresponding physical constraints, improves the physical consistency of the task processing result, and thus improves the accuracy of the task processing result. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are merely embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without any creative work.

[0012] Figure 1 A flowchart of a task processing method provided in an embodiment of the present application.

[0013] Figure 2 A structural diagram of a task processing model provided in an embodiment of the present application.

[0014] Figure 3 This is an architecture diagram of an autoencoder provided in an embodiment of the present application.

[0015] Figure 4 A flowchart of the training process of the task processing model provided in an embodiment of the present application.

[0016] Figure 5 This is a flowchart of the process of obtaining training samples provided in an embodiment of the present application.

[0017] Figure 6 A flowchart of the feature value search process provided in an embodiment of the present application.

[0018] Figure 7 A schematic diagram of the structure of a task processing device provided in an embodiment of the present application.

[0019] Figure 8 A schematic diagram of the structure of an electronic device provided in an embodiment of the present application. DETAILED DESCRIPTION

[0020] The technical solutions provided in the embodiments of the present application can be exemplarily applied to hardware devices such as processors, electronic devices, and servers (including cloud servers), or packaged into software programs and run. When the hardware devices execute the processing of the technical solutions in the embodiments of the present application, or the above-mentioned software programs are run, the automatic splitting of target tasks and the automatic calling of the application program interfaces required for the tasks can be achieved, thereby completing the purpose of the target tasks. The embodiments of the present application only provide an illustrative introduction to the specific processing of the technical solutions in the present application, and do not limit the specific implementation form of the technical solutions in the present application. Any technical implementation form that can execute the processing of the technical solutions in the present application can be adopted by the embodiments of the present application.

[0021] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0022] Before introducing this application solution, we first introduce the relevant technologies:

[0023] In the field of physical system modeling and engineering simulation, tasks such as structural vibration analysis, heat conduction simulation, electromagnetic wave propagation modeling, and quantum state solution often require modeling and analysis of the system's intrinsic characteristics, such as the structure's natural frequency and mode shapes, temperature field distribution patterns, electromagnetic wave propagation characteristics, or quantum state energy levels. Such tasks can be mathematically modeled using a class of self-adjoint second-order linear differential equations, namely the Sturm-Liouville equations, which are then converted into eigenvalue problems for solution. This type of modeling approach effectively captures the system's key dynamic behavior parameters, providing theoretical support for subsequent performance evaluation, condition monitoring, fault diagnosis, and optimal design.

[0024] Currently, there are several widely used solutions to the Sturm-Liouville problem involved in the above physical modeling tasks:

[0025] (1) Analytical solution: This method is applicable to problems of specific forms, such as Bessel functions and Legendre polynomials. By constructing analytical solutions, accurate eigenvalues ​​and eigenfunctions can be obtained under simple boundary conditions and constant coefficients. However, when faced with complex boundary conditions, nonlinear terms, or problems with variable coefficients, the computational complexity is high and analytical solutions are difficult to obtain.

[0026] (2) Traditional numerical methods: including finite difference method, Galerkin method, finite element method and spectral method. Finite difference method discretizes the differential operator and transforms the continuous problem into an algebraic problem. It can be applied to a variety of boundary conditions, but it has problems such as large computational complexity and limited accuracy in high-dimensional space and complex boundary conditions. Galerkin method and finite element method can adapt to irregular boundaries and complex geometric shapes by constructing trial functions to approximate characteristic functions. They are widely used in engineering simulation. However, they rely on fine grid division and a large amount of computing resources, making it difficult to solve nonlinear problems. Spectral method uses global functions (such as Fourier series or orthogonal polynomials) for approximation. It has high-order accuracy, but is sensitive to boundary conditions and has difficulty in handling irregular regions and complex boundary conditions.

[0027] (3) Machine learning methods: With the development of artificial intelligence, data-driven machine learning methods have gradually been applied to solving the Sturm-Liouville problem. However, these methods generally face problems such as limited generalization ability outside the training set, difficulty in effectively integrating physical prior knowledge, and insufficient numerical stability. As a result, the obtained solutions lack sufficient verifiability and reliability in engineering calculations.

[0028] In summary, all current solution methods have limitations when it comes to complex physical modeling tasks. Analytical methods are limited in their scope of application, traditional numerical methods are computationally expensive and lack adaptability, and machine learning methods suffer from significant shortcomings in integrating physical prior knowledge, numerical stability, and generalization across operating conditions. Therefore, a new modeling and solution technique is urgently needed that can achieve efficient and stable solutions to the Sturm-Liouville problem while ensuring physical consistency, thereby meeting the demand for accurate analysis of the intrinsic characteristics of the system in physical system modeling tasks.

[0029] In view of this, embodiments of the present application are directed to providing a task processing method, apparatus, device, medium, and product. This method, when processing Sturm-Liouville-related modeling tasks, introduces physical laws into the solution process to ensure that the obtained characteristic parameters adhere to the physical laws corresponding to the target task, thereby enhancing the physical consistency of the task processing results and improving the accuracy and reliability of the task processing results. These are described in detail in the following embodiments.

[0030] Exemplary Methods

[0031] Figure 1 This is a flowchart of a task processing method provided in an embodiment of the present application. Figure 1 As shown, the task processing method provided in this embodiment includes step S101 and step S102:

[0032] S101. Obtain modeling parameters required for modeling the target task using the Sturm-Liouville equation.

[0033] In this embodiment, the target task can be understood as a physical modeling task that involves mathematically modeling a second-order linear differential equation, namely the Sturm-Liouville equation, and then converting it into an eigenvalue problem for solution. The target task can include at least one of elastic modal analysis, heat conduction analysis, electromagnetic wave propagation characteristics analysis, and particle state analysis.

[0034] Elastic modal analysis tasks aim to determine the natural frequencies and mode shapes of a structure when subjected to dynamic loads. This helps understand how the structure vibrates and at what frequencies resonance may occur, thereby avoiding operational issues or failures caused by design flaws. For example, in the automotive industry, elastic modal analysis is performed on engine mounts to ensure that they do not resonate during operation due to matching the engine's vibration frequency, thereby reducing noise and vibration and improving durability.

[0035] Thermal conduction analysis studies how heat is transferred within a material or between different materials, helping to predict temperature distribution, optimize heat dissipation design, and prevent overheating. For example, in electronic device design, thermal conduction analysis can assess the effectiveness of heat sink design, ensuring that chips operate within a safe temperature range and preventing performance degradation or damage due to overheating.

[0036] The electromagnetic wave propagation characteristics analysis task aims to analyze the propagation behavior of electromagnetic waves (including radio waves, microwaves, infrared rays, etc.) in space, including phenomena such as reflection, refraction, and diffraction. This task is crucial for the design and optimization of wireless communication systems. For example, in the planning of mobile communication base stations, electromagnetic wave propagation characteristics analysis can be used to determine the optimal antenna position and orientation, ensuring signal coverage and service quality while minimizing interference.

[0037] Particle state analysis tasks are primarily used in the field of quantum mechanics. They investigate the states of particles at the microscopic scale (such as position, momentum, and energy). Solving the Schrödinger equation helps understand the fundamental properties and interactions of matter. For example, in semiconductor physics, particle state analysis can be used to calculate the energy level structure of electrons in crystals, thereby aiding the design of high-efficiency transistors and other electronic components. For example, by analyzing the electronic states in silicon or other semiconductor materials, new devices with specific electrical properties can be designed.

[0038] Modeling parameters refer to the input parameters required to solve the Sturm-Liouville equation, including boundary conditions, weight functions, potential functions, and operator coefficients. These input parameters collectively define the target task to be solved.

[0039] Boundary conditions define the constraints that the solution to the Sturm-Liouville problem must satisfy at the edges of a spatial region, such as Dirichlet, Neumann, and mixed boundary conditions. The physical meaning of boundary conditions varies in different application scenarios. For example, in a fixed string vibration analysis task, the boundary condition requires that the displacement at the boundary point be zero. In a heat conduction analysis task, the boundary condition requires that the temperature at the boundary be constant or adiabatic.

[0040] The weight function is introduced when defining the inner product. It is used to emphasize the contributions of different regions of the solution over a certain interval. It determines the measure of orthogonality between characteristic functions and can include constant weights, linear weights, and other methods. Physically, it can be interpreted as a weighting of a dielectric property or distribution, such as mass density, capacitance distribution, or thermal conductivity.

[0041] Potential functions typically reflect distributed constraints within a system or the influence of external fields. They can take the form of constants, periodic functions, or exponential potentials. In quantum mechanics, they correspond to the potential energy distribution of a particle. In elastic modal analysis, they represent the spatial variation of the stiffness or elastic modulus, influencing the solution shape and eigenvalue distribution.

[0042] Operator parameters describe the internal structure of differential operators and embody fundamental laws such as energy propagation, momentum transfer, and wave characteristics within the system. These parameters include the operator's order and constant parameters. In heterogeneous media, they reflect the spatial variation of material properties, such as thermal conductivity and stiffness coefficient.

[0043] In summary, when the target task includes elastic modal analysis, the boundary conditions include the displacement at both ends of the elastic object being zero; the weight function includes the density distribution of the elastic object; the potential function includes the tension of the elastic object and the equivalent potential energy distribution under the action of the external restoring force; and the operator parameters include the ratio of the mass per unit length of the elastic object to the tension.

[0044] When the target task includes a heat conduction analysis task, the boundary conditions include that the boundary of the object maintains a constant temperature or an adiabatic state; the weight function includes the heat capacity distribution of the object material; the potential function includes the spatial variation of thermal conductivity; and the operator parameters include at least one of thermal conductivity, density, and specific heat capacity.

[0045] When the target task includes an electromagnetic wave propagation characteristic analysis task, the boundary conditions include the interface conditions of the propagation medium; the weight function includes the magnetic permeability; the potential function includes the dielectric constant; and the operator parameter includes the square of the wave number in vacuum.

[0046] When the target task includes a particle state distribution task, the boundary conditions are determined according to the application scenario; the weight function includes the mass density distribution; the potential function includes the potential energy distribution of the particle's environment; and the operator parameters include the Planck constant and / or the particle mass.

[0047] S102. Solve the characteristic parameters of the Sturm-Liouville equation according to the modeling parameters and the physical laws corresponding to the target task to obtain the task processing result.

[0048] Physical laws refer to the a priori knowledge of physics that applies to specific mission scenarios. While physical laws manifest themselves in different forms across different target tasks, they all essentially constrain the behavior and state of a system.

[0049] Eigenvalues ​​are used to characterize the solution objectives corresponding to the target task, which satisfy the corresponding physical laws. Eigenvalues ​​include eigenvalues ​​and eigenfunctions. Eigenvalues ​​correspond to the eigenvalues ​​of the Sturm-Liouville problem, representing key physical quantities such as the system's "natural frequency" or "eigenenergy," reflecting the modal properties that the system can maintain in the absence of external perturbations. Each eigenvalue corresponds to an eigenstate of the system and is a key quantity for measuring the structural stability and response characteristics of the system.

[0050] The eigenfunction is the eigenstate of the system corresponding to the eigenvalue, which describes the spatial distribution structure of the system in a specific mode, and not only satisfies the Sturm-Liouville equation itself, but also needs to satisfy the given boundary conditions. For example, in the elastic modal analysis task, the eigenfunction represents the shape of the vibration mode; in the heat conduction analysis task, it describes the stable temperature distribution mode; and in quantum mechanics, it represents the wave function describing the state of a particle.

[0051] In the process of solving the Sturm-Liouville equation, the eigenvalue and the eigenfunction need to comply with the physical law corresponding to the target task. In different target tasks, the eigenvalue, the eigenfunction, and the physical law have different physical meanings. The physical meanings of the eigenvalue, the eigenfunction, and the physical law in different target tasks will be introduced as follows:

[0052] In the case where the target task includes an elastic modal analysis task, the eigenvalue includes the square of the vibration frequency; the eigenfunction includes the vibration modal shape of the elastic object at the vibration frequency; and the physical law includes at least one of energy conservation, momentum balance, material uniformity assumption, system linear small vibration assumption, solution continuity, and solution differentiability;

[0053] In the case where the target task includes a heat conduction analysis task, the eigenvalue includes the thermal diffusivity; the eigenfunction includes the temperature distribution mode in the steady state; and the physical law includes at least one of Fourier's heat conduction law, energy conservation, material property constraints, solution continuity, and solution differentiability;

[0054] In the case where the target task includes an electromagnetic wave propagation characteristic analysis task, the eigenvalue includes the propagation constant; the eigenfunction includes the distribution of the magnetic field in the medium; and the physical law includes at least one of the Maxwell equations that electromagnetic waves need to satisfy, the boundary conditions of the medium, the continuity of the solution, and the differentiability of the solution;

[0055] In the case where the target task includes a particle state distribution task, the eigenvalue includes the energy eigenvalue of the particle; the eigenfunction includes the wave function form under the energy eigenvalue; and the physical law includes at least one of the Schrödinger equation, the probability conservation principle, the continuity of the solution, and the differentiability of the solution.

[0056] In this embodiment, by introducing the physical law into the solving process when processing the Sturm-Liouville related modeling task, it is ensured that the obtained characteristic parameters comply with the physical law corresponding to the target task, so as to enhance the physical consistency of the task processing result, and improve the accuracy and reliability of the task processing result.

[0057] In some embodiments, a machine learning technique can be employed to solve a target task modeled based on a Sturm-Liouville problem. Specifically, after obtaining modeling parameters, the modeling parameters can be input to a task processing model to solve the Sturm-Liouville equation according to the modeling parameters and physical laws through the task processing model to obtain a task processing result. The task processing model can be trained according to modeling parameter samples corresponding to the Sturm-Liouville equation and a loss function. The loss function includes a physical information error loss for constraining the predicted characteristic parameters to satisfy the physical laws under the target task. The predicted characteristic parameters are obtained by solving the Sturm-Liouville equation according to the modeling parameter samples.

[0058] By introducing the physical information error loss into the loss function during the training process of the task processing model, it is ensured that the predicted characteristic parameters follow the physical laws corresponding to the target task, which not only enhances the physical consistency of the results of the task processing model in solving the Sturm-Liouville related modeling task, but also improves the accuracy and reliability of the task processing result. The implementation process of the task processing method of the embodiments of the present application will be described in detail below in combination with the architecture of the task processing model:

[0059] Figure 2 A structural diagram of a task processing model provided by the embodiments of the present application is shown in FIG. 1. As shown in FIG. 1, the task processing model includes an autoencoder (AE) 21, a physics-informed neural network (PINN) 22, and a Transformer network 23. Figure 2

[0060] The autoencoder 21 can be a variational autoencoder (VAE) for feature extraction and characteristic parameter prediction according to the input modeling parameters to generate initial characteristic parameters corresponding to the modeling parameters. The initial characteristic parameters include initial characteristic values and an initial characteristic function.

[0061] The physics-informed neural network 22 is used to adjust the initial predicted characteristic parameters according to the physical laws to obtain intermediate characteristic parameters. The intermediate characteristic parameters include intermediate characteristic values and an intermediate characteristic function.

[0062] The Transformer network 23 is used to model the intermediate characteristic function over a long distance to obtain the required characteristic function, and derive the required characteristic values by solving the characteristic function.

[0063] ​This embodiment captures long-distance features in the feature function through the Transformer network, which can improve the global consistency between the feature value and the feature function.

[0064] Based on the above network structure, in some embodiments, a specific implementation method for step S102 can be provided, including: generating initial eigenvalues ​​and initial eigenfunctions corresponding to the modeling parameters according to the modeling parameters through an autoencoder; adjusting the initial eigenfunction according to physical laws through a physical information neural network to obtain an intermediate eigenfunction; performing long-distance modeling on the intermediate eigenfunction through a Transformer network to obtain a eigenfunction, and using the initial eigenvalues ​​as eigenvalues.

[0065] The following will introduce the process of generating initial feature parameters by the autoencoder in conjunction with the architecture diagram of the autoencoder.

[0066] Figure 3 This is a diagram of the architecture of an autoencoder provided in an embodiment of the present application. Figure 3 As shown, the autoencoder 21 includes an encoder 211 and a decoder 212. The encoder 211 is used to map the modeling parameters to a low-dimensional latent space to obtain latent variables in the latent space to reduce computational complexity. The decoder 212 is used to generate initial feature parameters based on the latent variables in the latent space.

[0067] based on Figure 3 The architecture of the autoencoder shown generates initial feature parameters corresponding to the modeling parameters based on the modeling parameters through the autoencoder, including: mapping the modeling parameters to a low-dimensional latent space through the encoder to obtain latent variables in the latent space; and generating initial eigenvalues ​​and initial feature functions based on the latent variables in the latent space through the decoder.

[0068] Among them, the encoder maps the modeling parameters to a low-dimensional latent space to capture the nonlinear relationship between eigenvalues ​​and eigenfunctions, and reduces the data dimension, thereby reducing the computational complexity; the decoder generates the initial feature function based on the latent variables mapped to the latent space and derives the corresponding initial eigenvalues.

[0069] The physical information neural network 22 then adjusts the initial feature parameters according to actual physical laws to obtain intermediate feature parameters. Specifically, the neural network introduces physical equations related to the target task as regularization constraints. By adding the corresponding partial differential equation residual term to the loss function, the obtained feature parameters are ensured to meet the constraints of known physical laws while conforming to data-driven results. The intermediate feature parameters thus obtained have stronger physical consistency and generalization capabilities.

[0070] Finally, the Transformer network23 is used to model long-range dependencies between the intermediate feature functions within the intermediate feature parameters. Specifically, the Transformer's multi-head attention mechanism models the global interactions between feature functions, improving the model's adaptability to complex physical modeling tasks. After Transformer processing, an optimized feature function is obtained.

[0071] This embodiment combines the variational autoencoder (VAE), the physical information neural network (PINN), and the Transformer mechanism to ensure that the model can not only accurately capture the mapping relationship between eigenvalues ​​and eigenfunctions and achieve end-to-end mapping of Sturm-Liouville feature pairs, but also comply with physical constraints, thereby improving the accuracy and stability of the model.

[0072] In some embodiments, the neural network hierarchy of the task processing model may include an input layer, hidden layers, and an output layer; the hidden layers include convolutional layers, fully connected layers, and Transformer layers. The input layer performs preprocessing on the modeling parameters, such as formatting and normalization, and converts them into a high-dimensional vector representation recognizable by the network through an embedding mechanism. The convolutional layer, a spatial local feature modeling module, primarily extracts the local variation patterns of feature functions in the spatial domain. It can learn how changes in potential functions or weight functions within a certain region affect the shape of the feature function. The fully connected layer, a parameter interaction and nonlinear mapping mechanism, primarily captures the complex interactions and nonlinear effects between input parameters and intermediate features. Its function is similar to the expressiveness enhancement layer of a traditional neural network, mapping high-dimensional features to low-dimensional latent representations or the final output form. The Transformer layer, a global dependency modeling mechanism, enhances the understanding of cross-regional feature coupling. In particular, when feature functions exhibit "non-local" behavior within the entire domain (such as jumps or global symmetry), the Transformer can compensate for the shortcomings of CNNs in modeling long-range dependencies. It is a key module for modeling the global behavior of the system. The output layer, a feature extraction and result reconstruction module, outputs eigenvalues ​​and eigenfunctions. The eigenfunctions are typically reduced to a function distribution on the same scale as the input space through a decoder module. The eigenvalues, representing global scalar outputs, are derived by regression or derivation from the previous deep representations. The output layer is also key to reducing neural network results to physically meaningful solutions and is directly related to physical interpretability.

[0073] The following is a detailed introduction to the training process of the task processing model:

[0074] Figure 4 This is a flowchart of the training process of the task processing model provided in the embodiment of the present application. Figure 4As shown, the training process of the task processing model includes the following steps S401-S403:

[0075] S401. Solve the Sturm-Liouville equation based on the modeling parameter samples through a deep learning network to obtain prediction feature parameters.

[0076] The modeling parameter samples are a set of input parameters used to train the task processing model. Each modeling parameter sample includes a boundary condition sample, a weight function sample, a potential function sample, and an operator parameter sample.

[0077] Accordingly, each modeling parameter sample corresponds to real characteristic parameters, including real eigenvalues ​​and real characteristic functions. These real characteristic parameters are reference results obtained by accurately solving the corresponding Sturm-Liouville equation or performing high-precision numerical calculations. They serve as training labels in the supervised learning process, used to evaluate the accuracy of the model's predictions and guide the optimization direction of the model.

[0078] Deep learning methods rely on data for training, and high-quality training data helps improve training results. Therefore, before training, it is necessary to collect training samples, that is, modeling parameter samples. The following describes the process of obtaining training samples:

[0079] Figure 5 This is a flow chart of the process of obtaining training samples provided in the embodiment of this application. Figure 5 As shown, obtaining a modeling parameter sample includes the following steps S501-S504:

[0080] S501. Obtain characteristic parameters of a Sturm-Liouville problem with a known analytical solution under a first modeling parameter sample.

[0081] In some embodiments, step S501 includes: selecting a typical Sturm-Liouville problem with an analytical solution; solving the selected problem using a high-precision symbolic computing tool to obtain its eigenvalues ​​and eigenfunctions; and repeating the above steps under different boundary conditions to generate diverse training samples.

[0082] Specifically, high-precision symbolic computing tools (such as Mathematica or SymPy) can be used to solve typical Sturm-Liouville problems with analytical solutions, such as Bessel operators and Legendre operators, to extract their standard eigenvalues ​​(λ) and corresponding characteristic functions (y(x)), and obtain the characteristic parameters under the first modeling parameter sample.

[0083] To increase the diversity of training samples, the above solution process can also be performed under different boundary conditions. Different boundary conditions can not only reflect the needs of different physical scenarios, but also enrich the training samples.

[0084] Through step S501 , an eigenvalue and a eigenfunction of the Sturm-Liouville problem with a known analytical solution can be obtained as a basis for training sample data.

[0085] S502: Solve the Sturm-Liouville problem that does not have a known analytical solution using a numerical solution method to obtain characteristic parameters of the Sturm-Liouville problem that does not have a known analytical solution under the second modeling parameter sample.

[0086] In step S502, for complex Sturm-Liouville problems that do not have analytical solutions, numerical methods can be used to calculate eigenvalues ​​and eigenfunctions. For example, numerical methods such as spectral methods, finite element methods, or finite difference methods can be used to calculate the eigenvalues ​​and eigenfunctions of a Sturm-Liouville problem with complex boundary conditions and potential functions to obtain calculation results.

[0087] After completing the preliminary numerical calculations, the calculation results can be interpolated based on the pre-set discretization accuracy to ensure that the final eigenvalues ​​and characteristic functions maintain both high calculation accuracy and good numerical stability, thereby ensuring the smoothness and continuity of the data.

[0088] Through step S502, numerical solution data can be obtained as training samples. On this basis, in order to further expand the training set, the training samples can also be enriched by parameterized transformation methods. For details, please refer to the following introduction to step S503:

[0089] S503 : Construct a new modeling parameter sample by changing the boundary conditions, potential function, and weight function in the first modeling parameter sample and the second modeling parameter sample.

[0090] In order to enhance the generalization ability of the model and cover a wider range of physical scenarios, diversified training samples can be constructed by adjusting the boundary conditions, potential functions, and weight functions in the first modeling parameter sample and the second modeling parameter sample to enhance the generalization ability of the neural network.

[0091] Among them, changing the boundary conditions includes: randomly selecting different types of boundary conditions, such as Dirichlet boundary conditions, Neumann boundary conditions, and mixed boundary conditions, etc., and collecting multiple data points for each boundary condition type to ensure diverse coverage of boundary conditions.

[0092] Adjust the potential function and weight function, including: randomly generating a series of potential functions (such as constant potential, exponential potential, periodic potential, etc.) and weight functions (such as constant weight, linear weight, nonlinear weight, etc.), and designing transformation parameters to ensure that the generated training samples can cover tasks in various physical scenarios.

[0093] In some embodiments, after step S503, the method further includes preprocessing the first modeling parameter sample and the corresponding feature parameter, as well as the second modeling parameter sample and the corresponding feature parameter. The preprocessing step includes eigenvalue normalization and feature function orthogonal normalization to improve data quality and enhance the learning effect of the neural network.

[0094] Eigenvalue normalization involves mapping eigenvalues ​​to a uniform range, such as [0, 1] or [-1, 1]. This eliminates the effects of dimensional or magnitude differences between tasks and ensures stability and convergence efficiency during model training. Normalization methods can be flexibly selected based on the actual distribution of eigenvalues, such as z-score normalization or min-max normalization.

[0095] To satisfy the orthogonality requirement of the Sturm-Liouville theory for characteristic functions and help neural networks better learn their inherent structural properties, the numerically calculated characteristic functions must be orthogonally normalized. If the original characteristic functions fail to strictly satisfy the orthogonality constraint, regularization methods (such as Gram-Schmidt orthogonalization) can be used to adjust them. Furthermore, an adaptive weighting mechanism can be introduced to dynamically adjust the influence of the orthogonality constraint during training, thereby guiding the neural network to more effectively learn characteristic function representations that exhibit orthogonal properties.

[0096] S504: Determine the characteristic parameters under the new modeling parameter sample, and obtain the modeling parameter sample and the corresponding real characteristic parameters based on the first modeling parameter sample and the corresponding characteristic parameters, the second modeling parameter sample and the corresponding characteristic parameters, and the new modeling parameter sample and the corresponding characteristic parameters.

[0097] After preprocessing, in order to further enrich the training sample set, new modeling parameter samples can be determined to obtain additional feature parameters. This process includes strategies such as data perturbation and random parameter changes.

[0098] Data perturbation involves random perturbation of the characteristic function y(x), including operations such as adding noise, translation, or scaling, so as to simulate measurement errors or noise that may occur in actual applications, enhance the model's ability to handle uncertainty, and improve the model's robustness.

[0099] The random change of parameters includes random adjustment of the potential function parameters, weight function parameters, etc. in the Sturm-Liouville problem, and recalculation of the corresponding eigenvalues and eigenfunctions. Each parameter adjustment needs to ensure that the generated characteristic pair is consistent with the original problem, so that the neural network can learn the common characteristics under different parameter configurations, and improve the adaptability of the model to various parameter settings.

[0100] Next, in order to ensure that the final training sample set has sufficient representativeness and accuracy, and can effectively support the training of the neural network. The generated data set can also be cross-validated and physically constrained. Specifically, cross-validation can be used to verify whether the data set can stably reflect the essential characteristics of the Sturm-Liouville problem under different boundary conditions, potential functions and weight functions; and based on certain physical constraints (such as the distribution of eigenvalues, the orthogonality of eigenfunctions, etc.), the data set is filtered to remove data points that do not meet the requirements, ensuring the high quality of the data set.

[0101] Finally, before using all the characteristic pair data obtained by analytical and numerical solution methods, and the characteristic parameters under the new modeling parameter samples for neural network training, it needs to be converted into a data format suitable for neural network training, such as HDF5 or TFRecord, and ensure that the size of the data set is moderate, which can cover the diversity of Sturm-Liouville problems, and will not cause memory overflow or overfitting in the training process.

[0102] Through the above steps, a high-quality Sturm-Liouville characteristic pair data set can be constructed, which contains several modeling parameter samples and corresponding real characteristic parameters.

[0103] After obtaining the modeling parameter samples and corresponding real characteristic parameters through the above steps S501 to S504, the deep learning network can be trained using them to obtain a task processing model. And by inputting the modeling parameter samples into the task processing model, the solution of the Sturm-Liouville equation is predicted, thereby generating the corresponding predicted characteristic parameters, i.e. predicted eigenvalues and predicted eigenfunctions. The architecture of the deep learning network can refer to the model architecture shown in Figure 2 In some embodiments, the step S401 of solving the Sturm-Liouville equation according to the modeling parameter samples by the deep learning network to obtain the predicted characteristic parameters includes the following steps a1-a3:

[0104] Step a1, generating the initial predicted eigenvalues and initial predicted eigenfunctions corresponding to the modeling parameter samples by the autoencoder according to the modeling parameter samples.

[0105] The step a1 comprises: mapping the modeling parameter sample to a low-dimensional latent space by an encoder to obtain a hidden variable in the latent space; and generating an initial predicted characteristic value and an initial predicted characteristic function according to the hidden variable in the latent space by a decoder.

[0106] The encoder maps the modeling parameter sample to the low-dimensional latent space to capture the nonlinear relationship between the characteristic value and the characteristic function, and to reduce the data dimension, thereby reducing the computational complexity; and the decoder generates the initial predicted characteristic function according to the hidden variable mapped to the latent space, and derives the corresponding initial predicted characteristic value.

[0107] The step a2 comprises: adjusting the initial predicted characteristic function according to a physical law by a physical information neural network to obtain an intermediate predicted characteristic function.

[0108] The physical information neural network 22 adjusts the initial predicted characteristic function according to the actual physical law to obtain the intermediate predicted characteristic function. Specifically, the physical equation related to the target task is introduced as a regularization constraint condition in the neural network, and the corresponding partial differential equation residual term, i.e., the physical information error loss, is added to the loss function, so as to ensure that the predicted characteristic function meets the data-driven result and satisfies the limitation of the known physical law. The intermediate predicted characteristic function obtained in this way has stronger physical consistency and generalization ability.

[0109] The step a3 comprises: long-distance modeling the intermediate predicted characteristic function by a Transformer network to obtain a predicted characteristic function, and taking the initial predicted characteristic value as a predicted characteristic value.

[0110] The Transformer network 23 is used to model the long-distance dependence relationship of the intermediate predicted characteristic function. Specifically, the global interaction relationship between the intermediate predicted characteristic functions is modeled by the multi-head attention mechanism in the Transformer, so as to capture more complex feature structures. After the processing of the Transformer, the optimized predicted characteristic function and the initial predicted characteristic value are taken as the predicted characteristic parameters together.

[0111] Continuing to refer to Figure 4 After the step S401, a step S402 can also be included.

[0112] S402, based on the difference between the predicted characteristic parameters and the real characteristic parameters, based on the verification result of the predicted characteristic parameters based on the boundary condition sample, and based on the verification result of the predicted characteristic parameters based on the physical law, a comprehensive loss value corresponding to a loss function is obtained.

[0113] During model training, a loss function guides the adjustment of model parameters to minimize the error between the predicted and true results. The loss function includes at least one of the following: feature parameter error loss, physical information error loss, boundary condition error loss, and orthogonality constraint loss.

[0114] Feature parameter error loss can be obtained by comparing the predicted feature parameters with the actual feature parameters. Feature parameter error loss includes eigenvalue error loss and feature function error loss. The eigenvalue error loss can be obtained by comparing the predicted feature value with the actual feature value; the feature function error loss can be obtained by comparing the predicted feature function with the actual feature function.

[0115] The physical information error loss is used to assess whether the predicted eigenvalues ​​and predicted eigenfunctions conform to established physical laws. This ensures that the model output is not only numerically close to the true value but also physically reasonable. The boundary condition error loss is calculated by verifying whether the predicted eigenvalues ​​and predicted eigenfunctions satisfy a given boundary condition sample. This loss ensures the accuracy of the solution. The orthogonality constraint loss is used to check whether different predicted eigenfunctions maintain the required orthogonality.

[0116] In some embodiments, step S402 obtains a comprehensive loss value corresponding to the loss function based on the difference between the predicted characteristic parameters and the actual characteristic parameters, the verification results of the predicted characteristic parameters based on the boundary condition samples, and the verification results of the predicted characteristic parameters based on the physical laws, including the following steps b1 to b5:

[0117] Step b1: determining the feature parameter error loss based on the difference between the predicted feature parameter and the true feature parameter.

[0118] The predicted feature parameters include predicted feature values ​​and predicted feature functions; the true feature parameters include true feature values ​​and true feature functions. Accordingly, the feature parameter error loss includes feature value error loss and feature function error loss. Step b1 then includes: determining the feature value error loss based on the difference between the predicted feature values ​​and the true feature values; and determining the feature function error loss based on the difference between the predicted feature functions and the true feature functions.

[0119] The eigenvalue error loss can be calculated using the mean squared error (MSE) or other regression loss functions. The eigenfunction error loss can also be calculated using the mean squared error (MSE) or other distance metrics in the function space (such as the L2 norm).

[0120] Step b2: Based on the verification result of whether the predicted characteristic value and the predicted characteristic function satisfy the physical law, the physical information error loss is obtained.

[0121] The physical information error loss is constructed by substituting the predicted eigenvalues ​​and predicted eigenfunctions into the Sturm-Liouville equation, calculating the residual term, and integrating or averaging the residual over the entire domain.

[0122] Step b3: Based on the verification results of whether the predicted characteristic values ​​and the predicted characteristic functions meet the boundary condition samples, the boundary condition error loss is obtained.

[0123] Specifically, it can be obtained by calculating the deviation of the predicted characteristic value and the predicted characteristic function at the boundary according to the boundary condition type.

[0124] Step b4: determining the orthogonality constraint loss based on the verification results of whether different prediction feature functions satisfy the orthogonality condition.

[0125] Specifically, for any two eigenfunctions of different orders, their orthogonality under the weighted inner product can be calculated to obtain the orthogonality constraint loss.

[0126] Step b5: Obtain a comprehensive loss value based on the weighted sum of the characteristic parameter error loss, the physical information error loss, the boundary condition error loss, and the orthogonality constraint loss.

[0127] Assume that the comprehensive loss function is recorded as L total , then L total It can be expressed as the following formula (1):

[0128] L total =w1*L λ +w2*L y +w3*L phys +w4*L bc +w5*L orth ; (1)

[0129] Where, L λ represents the eigenvalue error loss; L y Represents the characteristic function error loss; L phys Indicates the physical information error loss; L bc represents the boundary condition error loss; L orth represents the orthogonality constraint loss; w1, w2, w3, w4, and w5 are the weight coefficients corresponding to each loss.

[0130] In some embodiments, in order to ensure the effectiveness of the optimization process and improve the quality of the final solution, a dynamic adaptive strategy can be used to adjust the weight coefficient of the orthogonality constraint loss to avoid limiting the solution exploration space due to excessively strong orthogonality constraints in the early stages of optimization, or failing to ensure the necessary orthogonality between eigenfunctions due to excessively weak constraints in the later stages of optimization. At the same time, the weight of the physical information error loss can be set according to the accuracy requirements of the specific physical modeling task, where the accuracy requirements of the physical modeling task are positively correlated with the weight setting, that is, the higher the accuracy requirement, the greater the weight, to ensure that the model can accurately reflect the actual physical laws. Boundary condition error, eigenvalue error, and eigenfunction function are basic error terms, and their weights need to maintain a stable and appropriate proportion.

[0131] In some embodiments, the weight coefficients of various loss functions can be adjusted dynamically or in stages. For example, in the early stages of training, the weights of orthogonality constraints and physical information error loss can be appropriately reduced to give the algorithm greater freedom to explore possible solutions. As training progresses, the weights of these constraints can be gradually increased to ensure that the final solution meets both physical requirements and has good mathematical properties.

[0132] Furthermore, by monitoring key metrics such as error trends and convergence speed, the weights of various loss functions can be automatically adjusted, ensuring efficient and reliable training. For example, when the model begins to converge stably, orthogonality and physical information constraints are gradually strengthened to guide the model to achieve higher accuracy and stability.

[0133] This embodiment adjusts the weights of each loss function through a weight adjustment mechanism, which can not only improve the model's ability to solve complex problems, but also enhance its adaptability to different application scenarios, thereby obtaining a task processing model with strong robustness and high generalization ability, so as to improve the accuracy of task processing results.

[0134] Continue reading Figure 4 , after step S402, step S403 may also be included.

[0135] S403: Converging the deep learning network based on the comprehensive loss value to obtain a task processing model.

[0136] By minimizing the combined loss and adjusting the parameters of the deep learning network, the model can accurately predict the eigenvalues ​​and eigenfunctions of the Sturm-Liouville equation, ensuring that these predictions meet physical laws, boundary conditions, and orthogonality constraints. After sufficient training, a task processing model can be obtained.

[0137] To minimize the comprehensive loss value, the back propagation algorithm can be used to calculate the gradient of the loss function with respect to each network parameter, and the network parameters are updated using an optimization algorithm until the model converges. Among them, the optimization algorithm includes stochastic gradient descent (SGD), Adam (Adaptive Moment Estimation) optimizer, etc. The conditions for model convergence include: the comprehensive loss value is less than the preset loss threshold, the performance of the model on the validation set no longer improves significantly (early stopping method), the parameter update amount is small, or the number of iterations reaches the predetermined maximum number of iterations.

[0138] During the training process, appropriate training strategies such as activation functions, anti-overfitting strategies, weight initialization, self-supervised learning, and data augmentation can be used.

[0139] Among them, ReLU (Rectified Linear Unit) can be used as the main activation function to handle non-linear relationships, and a linear activation function can be used in the output layer to ensure that the output range of the eigenvalue λ and the eigenfunction y(x) is appropriate.

[0140] Anti-overfitting strategies can prevent the model from overfitting and improve the model's generalization ability. This strategy includes Dropout and L2 regularization methods.

[0141] During the training process, using the Xavier initialization method to initialize the weights can avoid gradient vanishing or explosion and promote rapid convergence in the early stages of training.

[0142] When the number of samples is insufficient, self-supervised learning can also be used with existing physical constraints (such as differential equation residuals), and data can be enhanced through perturbation of the eigenfunction or boundary conditions to improve model robustness.

[0143] To improve the convergence speed and accuracy of the model, the Adam optimizer can be used in the early stages of training to accelerate the convergence process, and the L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) optimizer can be used in the later stages for fine tuning to improve model accuracy.

[0144] To evaluate the generalization ability of the model, the test set can be used for error evaluation. For example, different types of Sturm-Liouville problems (such as Bessel operators, Schrödinger equations, etc.) can be used to test the model's performance, calculate the eigenvalue prediction error and eigenfunction prediction error, or observe the model's adaptability by changing the input parameters, or use Monte Carlo sampling to evaluate the model's stability under various boundary conditions.

[0145] In order to control the error of the model, the error regression model (Residual Learning) can be used in combination with adaptive regularization technology to dynamically adjust the network structure and the weights of various loss functions to further optimize the model prediction accuracy and reduce the risk of overfitting.

[0146] Specifically, the acquired complete dataset can be divided into a training set and a validation set. Model performance can be regularly evaluated on the validation set during training, while monitoring changes in the loss function to track training progress. Based on the validation results, the network structure and hyperparameters can be dynamically adjusted. For example, the number of network layers, the number of nodes per layer, and the type of activation function can be gradually adjusted based on the model's performance on the validation set, thereby modeling the mapping relationship between feature pairs.

[0147] Through the above training strategy, the model is able to handle the complex Sturm-Liouville feature pair problem solving task. Moreover, the model can not only meet the physical constraints, but also accurately predict the relationship between eigenvalues ​​and eigenfunctions, and has good generalization ability and robustness.

[0148] In the process of solving the Sturm-Liouville eigenvalue problem, an eigenvalue search is required to find the optimal solution that satisfies the Sturm-Liouville differential operator's eigenvalue conditions. During the eigenvalue search process, it is necessary to define an indicator to measure the solution error, which reflects the degree of deviation between the current eigenvalue and the true eigenvalue. By continuously optimizing this indicator, the true eigenvalue can be gradually approached. In other words, the eigenvalue search is guided by feedback information on the degree of deviation. To improve the efficiency of the eigenvalue search, this embodiment can combine reinforcement learning to perform the eigenvalue search. The following is a detailed introduction to the eigenvalue search process with the accompanying drawings:

[0149] Figure 6 This is a flow chart of the feature value search process provided in the embodiment of the present application. Figure 6 As shown, the feature value search process includes the following steps S601-S604:

[0150] S601: Determine the eigenvalue error loss based on the difference between the predicted eigenvalue and the true eigenvalue.

[0151] Among them, the eigenvalue error loss represents the degree of deviation between the current eigenvalue and the true eigenvalue.

[0152] S602: Perform eigenvalue search based on eigenvalue error loss and a preset eigenvalue error index by using a reinforcement learning method to obtain an adjustment step size and adjustment direction for the next eigenvalue search.

[0153] In step S602, the reinforcement learning agent (Agent) takes the current eigenvalue error loss and the preset target eigenvalue error as input and outputs the step size and direction for the next eigenvalue adjustment. Specifically, the agent tries different adjustment strategies and provides reward feedback based on the error improvement achieved after executing the strategy.

[0154] For example, after each strategy adjustment, the error in the eigenvalue after this adjustment is evaluated to see if it has decreased compared to the previous one. If the error decreases and the reduction is greater than or equal to a preset threshold, a positive reward is given; conversely, if the error increases or the reduction is less than the preset reduction threshold, a negative reward or a low reward is given. Through continuous iterative learning, reinforcement learning can gradually converge to the optimal adjustment strategy, thereby avoiding local minima and improving global search capabilities.

[0155] S603: Obtain the predicted eigenvalue corresponding to the next eigenvalue search according to the adjustment step size and adjustment direction of the next eigenvalue search and the predicted eigenvalue.

[0156] After obtaining the adjustment step size and adjustment direction output by the reinforcement learning strategy, the current predicted feature value can be updated along the adjustment direction according to the adjustment step size to generate the predicted feature value used in the next round of search.

[0157] In this way, the model can dynamically explore the feature space and gradually approach the true eigenvalues. This update mechanism ensures the continuity and controllability of the eigenvalue search process, while providing more instructive supervision signals for subsequent network training.

[0158] S604. Continue training the deep learning network based on the predicted eigenvalue corresponding to the next eigenvalue search, and return to the step of determining the eigenvalue error loss based on the difference between the predicted eigenvalue and the true eigenvalue, and determine the comprehensive loss value based on the eigenvalue error loss, converge the deep learning network based on the comprehensive loss value, and obtain the task processing model.

[0159] After the eigenvalue update is complete, the newly predicted eigenvalue is re-input into the deep learning network as one of the input conditions for a new round of training, and the model parameters are updated and optimized. Then, the operation in S601 is performed again, that is, the eigenvalue error loss is recalculated based on the difference between the latest prediction result and the true value.

[0160] On this basis, a comprehensive loss function is constructed based on the eigenvalue error loss and other possible task-related losses (such as physical consistency loss). By continuously iterating this process, the model gradually reduces the prediction error, improves generalization and stability, and ultimately achieves convergence of the deep learning network. The converged model is a task processing model that can be used to perform the target task, with high accuracy, robustness, and good physical consistency.

[0161] Preferably, convergence judgment conditions can also be set, such as when the eigenvalue error drops below a preset threshold or the error change in multiple consecutive training cycles is less than a set range, the model is judged to have converged and the training process ends.

[0162] In some embodiments, to avoid the local optimum problem that may occur in the reinforcement learning mechanism due to the complexity of the search space, a gradient-enhanced random search can also be combined with the eigenvalue search process to improve global convergence. Gradient-enhanced random search introduces directional guidance based on gradient information on the basic search path provided by reinforcement learning, and performs a certain range of random perturbations in the local area, thereby ensuring that the search path is both directional and retains a certain degree of exploration capability, thereby improving the overall optimization effect. Then, step S602, when performing an eigenvalue search based on the eigenvalue error loss and a preset eigenvalue error index using the reinforcement learning method to obtain the adjustment step size and adjustment direction for the next eigenvalue search, specifically includes: when the current search is in the first search stage, determining the adjustment step size for the next eigenvalue search as a first adjustment step size based on the eigenvalue error loss and the preset eigenvalue error index using the reinforcement learning method; when the current search is in the second search stage, determining the adjustment step size for the next eigenvalue search as a second adjustment step size based on the eigenvalue error loss and the eigenvalue error index using the reinforcement learning method; wherein the first adjustment step size is greater than the second adjustment step size, and the second search stage is located after the first search stage.

[0163] The first search phase can be understood as the early stages of the search, while the second phase corresponds to the later stages. In the early stages of training, the reinforcement learning strategy has not yet formed an effective policy mapping relationship and has not fully understood the entire search space. At this point, to guide it to effectively explore a wider parameter space and avoid prematurely falling into local optimal solutions, it is necessary to introduce a strong randomization mechanism, such as setting a large perturbation amplitude or a high exploration probability.

[0164] As training progresses, entering the second search phase, the reinforcement learning agent has accumulated a certain amount of experience, and the model error tends to converge or decreases significantly more slowly. At this point, the search strategy should shift to leveraging learned knowledge, reducing unnecessary random exploration, and relying on gradient information or experience-based decision-making to perform detailed parameter optimization to improve convergence efficiency and final model accuracy.

[0165] The switching between the above search phases can be achieved through a dynamic control mechanism. This mechanism can adaptively judge the current phase based on key indicators such as the slope change of the error descent curve, the changing trend of the cumulative reward, and the degree of convergence of the policy network. By adjusting policy parameters such as the exploration rate and perturbation amplitude, a balanced transition from exploration to utilization can be achieved, thereby ensuring the comprehensiveness of the search while improving the overall convergence efficiency.

[0166] In some examples, the first search phase and the second search phase can be dynamically determined based on the number of iterations, the error convergence trend, or the performance improvement rate during the training process.

[0167] In some embodiments, to ensure the stability of the search process and the reliability of the final solution, it is necessary to evaluate convergence and design a reasonable error control mechanism. Specifically, during the entire eigenvalue search process, the changing trend of the eigenvalue error is monitored in real time to determine whether the current search strategy is effective.

[0168] If the eigenvalue error doesn't decrease significantly over multiple iterations, it indicates that the current search strategy may be stuck in a local optimum or the current search step size is improperly set, resulting in reduced optimization efficiency. In this case, it's necessary to readjust the optimization parameters, such as expanding the random search range or adjusting the reinforcement learning reward mechanism, to guide the model out of its current inefficient state and continue to approach a better solution.

[0169] The specific implementation method of the above process is as follows: determine the reduction between the eigenvalue error loss of each iteration and the eigenvalue error loss of the previous iteration in the eigenvalue error loss of N consecutive iterations, and obtain the decrease values ​​of the N eigenvalue error losses; if the proportion of the decrease values ​​of the N eigenvalue error losses that are less than a preset decrease threshold exceeds a preset proportion, then adjust the optimization parameters of the reinforcement learning method, and the optimization parameters include the reward mechanism and / or the eigenvalue search range.

[0170] Through the above mechanism, it is possible to respond in a timely manner when the search efficiency decreases and dynamically adjust the optimization strategy, thereby improving the robustness and convergence ability of the overall algorithm.

[0171] In some embodiments, when processing a physical modeling task based on the Sturm-Liouville problem, in addition to accurately solving the eigenvalues, it is also necessary to generate a high-precision characteristic function. The calculation accuracy of the characteristic function will affect the quality of the solution to the Sturm-Liouville problem. Therefore, in the process of solving the characteristic function, it is necessary to set an adaptive error control mechanism to ensure the accuracy and numerical stability of the characteristic function. The error control mechanism monitors the error performance of the characteristic function in different areas in real time, and dynamically adjusts the corresponding error control strategy according to the error size. These strategies include local error monitoring strategy, multi-scale solution strategy and adaptive optimization step strategy.

[0172] Among them, the specific implementation method of the local error monitoring strategy includes: selecting multiple monitoring points in the calculation area of ​​the Sturm-Liouville equation, and determining the characteristic function error loss corresponding to the predicted characteristic function at each monitoring point. The calculation area is determined according to the boundary condition samples; according to the characteristic function error loss at each monitoring point, the error control strategy in the corresponding calculation area is adjusted.

[0173] The multi-scale solution strategy adopts a staged optimization method. First, a low-precision neural network is used for preliminary calculations to quickly obtain the approximate distribution of the characteristic function. Then, a high-precision algorithm is used to perform in-depth optimization on key areas that need to be refined, thereby achieving an organic combination of global overview and local details, and improving solution efficiency and accuracy.

[0174] The adaptive optimization step size strategy is implemented by dynamically adjusting the learning rate of the neural network during training. For areas with large errors, the learning rate is automatically increased to provide more optimization opportunities and accelerate convergence. For areas with small errors, the learning rate can be reduced or computing resources can be allocated less to avoid overfitting, effectively improving overall computing efficiency.

[0175] By introducing the above error control strategy, not only the calculation accuracy of the characteristic function is improved, but also the adaptability and flexibility in solving complex problems are increased.

[0176] In some embodiments, the stability and generalization ability of the characteristic function can ensure the reliability of the calculation results in practical applications. To verify the effectiveness of the solution process, after completing the characteristic function solution, it is necessary to verify its stability and generalization ability from multiple perspectives, including:

[0177] First, the same Sturm-Liouville problem is solved multiple times independently using different initialization conditions and network architectures, and the consistency of the characteristic functions obtained from each solution is determined. If the characteristic functions obtained under different initialization conditions show high consistency, it indicates that the method is stable and can effectively resist the influence of changes in initial conditions.

[0178] Secondly, to assess the generalizability of the method, it can be tested on a range of Sturm-Liouville systems with different parameter settings. For example, by varying the weight and potential functions, the algorithm can be verified to be adaptable to a variety of mathematical structures. If the calculated characteristic functions retain reasonable mathematical properties and meet the basic requirements of the Sturm-Liouville theory, the method demonstrates good generalization and applicability to a wide range of physical scenarios.

[0179] Finally, the calculated characteristic function is compared with the analytical solution or high-precision numerical solution to verify whether it meets the expected theoretical accuracy. This detailed error analysis not only confirms the accuracy of the model but also provides a basis for further optimization of the calculation process, ensuring that the final solution meets the requirements of practical applications.

[0180] In summary, the above-mentioned multi-faceted verification methods, including stability testing, generalization ability assessment, and comparative analysis with standard solutions, can not only comprehensively verify the quality of the characteristic function but also guide subsequent optimization work, making deep learning-based methods more reliable and practical in solving the Sturm-Liouville problem.

[0181] In some embodiments, in order to evaluate the effectiveness of the technical solution of the embodiment of the present application in solving the Sturm-Liouville eigenvalue problem, its accuracy, stability and computational efficiency can be fully verified through a series of systematic experiments, and its feasibility in practical applications can be further explored. This experimental evaluation not only includes standard numerical simulations, but also needs to be tested in combination with actual problems in specific engineering and physical backgrounds to fully demonstrate the adaptability and practicality of the method. The entire experimental process mainly includes the following four aspects: numerical experiment design, benchmark analysis, complex scenario application, and computational efficiency evaluation.

[0182] 1. Numerical Experiment Design

[0183] During the experimental phase, a set of representative Sturm-Liouville problems was constructed to test the solution capability of the proposed scheme under different conditions. The design of the experimental data should comprehensively consider the following key factors:

[0184] Diversity of boundary conditions: Select a variety of typical boundary conditions, such as Dirichlet, Neumann, and mixed boundary conditions, to evaluate the adaptability of the algorithm to different physical constraints.

[0185] Variations of weight functions and potential functions: Introduce weight functions and potential functions in different forms, such as constant, exponential growth, and periodic, to test their robustness in solving complex mathematical structures.

[0186] The impact of eigenvalue order: This includes verifying the computational accuracy of low-order eigenvalues ​​and evaluating the convergence behavior of high-order eigenvalues ​​to ensure that the method maintains good stability and accuracy in high-order modes.

[0187] Simulation of noise interference: Introducing a certain amplitude of random noise during the data generation process to simulate measurement errors or uncertainties in practical applications, thereby evaluating the algorithm's anti-interference ability under non-ideal conditions.

[0188] After completing the experimental design, the technical solution of the embodiment of the present application is used to solve the problem, and compared with traditional numerical methods (such as the finite difference method, the Galerkin method or the spectral method), and the advantages and limitations of this solution are comprehensively analyzed from the aspects of solution accuracy and convergence speed.

[0189] 2. Benchmark Analysis

[0190] To achieve quantitative evaluation, we can select several standard Sturm-Liouville problems with analytical or high-precision numerical solutions as benchmark test cases. These standard problems can effectively verify the accuracy and reliability of our solution. During testing, we should pay attention to the following performance indicators:

[0191] Relative error analysis: Calculate the relative error between the predicted eigenvalue and the true eigenvalue to evaluate the overall solution accuracy.

[0192] Convergence test: Determine the convergence trend of the algorithm under different training rounds or calculation steps to ensure that the results converge stably within a reasonable range.

[0193] Orthogonality verification: Check whether the obtained characteristic functions meet the basic requirements of the Sturm-Liouville theory on the orthogonality of characteristic functions, and verify the mathematical consistency of the method.

[0194] Computation time comparison: Under the same accuracy requirements, compare the computation time required for the technical solution of the embodiment of the present application with that of the traditional method to evaluate their potential for improving computational efficiency.

[0195] Through the above benchmark tests, the performance advantages of the technical solutions of the embodiments of the present application can be quantitatively identified, and a clear direction can be provided for subsequent optimization.

[0196] 3. Complex Scenario Application

[0197] The Sturm-Liouville problem is widely used in many fields, including physics, engineering, and signal processing. Therefore, it needs to be verified in more challenging real-world scenarios to demonstrate its broad applicability. The main application scenarios include:

[0198] Quantum mechanics problem: Solving the Schrödinger equation is essentially a classic Sturm-Liouville problem. This experiment will use different potential function models to test the performance of this scheme in quantum state energy calculations and compare it with high-precision numerical solutions.

[0199] Vibration modal analysis: In mechanical systems, the vibration modes of structures such as beams and plates can be modeled as Sturm-Liouville problems. This approach is tested for its applicability to modal analysis tasks by varying structural parameters and boundary conditions.

[0200] Signal processing applications: In Fourier transform and wavelet analysis, characteristic functions can be used for signal decomposition and filtering. This experiment will test the effectiveness of this solution in specific signal processing tasks.

[0201] Furthermore, it is necessary to test its robustness under different problem sizes, boundary conditions, or noise perturbations. If instability occurs under certain conditions, the generalization and stability of the method can be improved by adjusting the optimization strategy, introducing additional physical constraints, or refining the loss function design.

[0202] 4. Computational Efficiency Evaluation

[0203] In addition to accuracy and stability, computational efficiency is another key indicator of practicality. Especially in large-scale or real-time application scenarios, the time complexity of the numerical solution directly affects the feasibility of the method. To this end, the experiment focuses on evaluating the following aspects:

[0204] Computational time analysis: Record the running time of the technical solution of the embodiment of this application under different problem scales, and compare it with traditional numerical methods to analyze its efficiency advantages in large-scale problems.

[0205] Parallel computing capabilities: Since deep learning frameworks naturally support GPU acceleration, we can further test the execution efficiency of this solution on different computing devices (CPU vs. GPU) and evaluate its scalability in high-performance computing environments.

[0206] Multi-task solving capability: Test whether the technical solution of the embodiment of this application can share model parameters or computing resources among multiple related Sturm-Liouville problems, for example, by reducing the cost of repeated training through transfer learning or parameter reuse, thereby improving the overall solving efficiency.

[0207] High-dimensional problem adaptability: Evaluate the performance of the technical solutions of the embodiments of this application on high-dimensional Sturm-Liouville problems. If the technical solutions can maintain low computational complexity and high solution accuracy in high-dimensional space, it indicates that they have potential for promotion in a wider range of application scenarios.

[0208] By systematically conducting numerical experiments, benchmark tests, complex scenario applications, and computational efficiency evaluations, we can fully verify the accuracy, stability, and efficiency of the technical solutions of the embodiments of this application in solving the Sturm-Liouville eigenvalue problem.

[0209] In summary, the task processing method, apparatus, device, and product provided in the embodiments of the present application have the following beneficial effects:

[0210] (1) By combining a physical information neural network (PINN) with the Transformer architecture, they are used to solve the Sturm-Liouville problem. PINN can optimize the solution process of eigenvalues ​​and eigenfunctions by leveraging physical prior knowledge, while the Transformer network, through its powerful sequence modeling capabilities, enhances the learning and expression capabilities of complex boundary conditions and differential operators. This combination can improve solution accuracy and generalization capabilities, and is particularly suitable for Sturm-Liouville problems with complex boundary conditions and nonlinear coefficients.

[0211] (2) Using VAE as the eigenvalue search module, through an effective encoding and decoding process, it can quickly mine the most suitable eigenvalues ​​from a vast search space. VAE uses the learning ability of the latent space to more efficiently find the optimal solution to physical problems in high-dimensional space. Compared with traditional numerical methods, it can reduce computational overhead and optimize convergence issues.

[0212] (3) By introducing reinforcement learning (RL) into the eigenvalue calculation process, the network structure and hyperparameters are automatically adjusted to optimize solution accuracy and computational efficiency. The dynamic adaptability of the RL algorithm enables the model to have dynamic adaptability. It can flexibly adjust the search direction and step size according to the current error state, avoiding falling into local optimality, and improving the robustness and generalization performance of the algorithm under different problem settings.

[0213] (4) By constraining neural network training through the physical guidance loss function, the model can be continuously guided to better follow the physical constraints during the neural network training process, avoiding the generation of solutions that do not conform to physical meaning. Through this constraint, the model can fully consider the physical characteristics of the problem during the learning process and ensure the consistency between the numerical solution and the physical solution.

[0214] Exemplary devices

[0215] Corresponding to the above-mentioned task processing method, an embodiment of the present application also provides a task processing device. Figure 7 This is a structural diagram of a task processing device provided in an embodiment of the present application. Figure 7 As shown, the task processing device provided by the embodiment of the present application includes: a task processing unit 701; wherein the task processing unit 701 is used to perform the following steps: obtaining the modeling parameters required for modeling the target task through the Sturm-Liouville equation, and the target task includes at least one of the elastic modal analysis task, the heat conduction analysis task, the electromagnetic wave propagation characteristic analysis task and the particle state analysis task; solving the characteristic parameters of the Sturm-Liouville equation according to the modeling parameters and the physical laws corresponding to the target task to obtain the task processing result, the characteristic parameters characterizing the solution target corresponding to the target task, and the solution target satisfies the physical laws.

[0216] In some embodiments, the task processing method is executed through a task processing model, which includes an autoencoder, a physical information neural network and a Transformer network, and the characteristic parameters include eigenvalues ​​and characteristic functions; wherein the task processing unit 701 solves the characteristic parameters of the Sturm-Liouville equation according to the modeling parameters and the physical laws corresponding to the target task, and obtains the task processing results, specifically including: generating the initial eigenvalues ​​and initial eigenfunctions corresponding to the modeling parameters according to the modeling parameters through the autoencoder; adjusting the initial eigenfunction according to the physical laws through the physical information neural network to obtain an intermediate eigenfunction; performing long-distance modeling on the intermediate eigenfunction through the Transformer network to obtain the eigenfunction, and using the initial eigenvalues ​​as the eigenvalues.

[0217] In some embodiments, the autoencoder includes an encoder and a decoder; wherein, the task processing unit 701 generates initial feature parameters corresponding to the modeling parameters based on the modeling parameters through the autoencoder, including: mapping the modeling parameters to a low-dimensional latent space through the encoder to obtain latent variables in the latent space; generating the initial eigenvalues ​​and the initial feature functions based on the latent variables in the latent space through the decoder.

[0218] In some embodiments, the task processing model is trained using the following training steps: solving the Sturm-Liouville equation according to the modeling parameter samples through a deep learning network to obtain predicted feature parameters; the modeling parameter samples correspond to real feature parameters, and the modeling parameter samples include boundary condition samples; based on the difference between the predicted feature parameters and the real feature parameters, the verification results of the predicted feature parameters based on the boundary condition samples, and the verification results of the predicted feature parameters based on the physical laws, the comprehensive loss value corresponding to the loss function is obtained; based on the comprehensive loss value, the deep learning network is converged to obtain the task processing model.

[0219] In some embodiments, the comprehensive loss value corresponding to the loss function is obtained based on the difference between the predicted characteristic parameters and the actual characteristic parameters, the verification results of the predicted characteristic parameters based on the boundary condition samples, and the verification results of the predicted characteristic parameters based on the physical laws, including: determining the characteristic parameter error loss based on the difference between the predicted characteristic parameters and the actual characteristic parameters; obtaining the physical information error loss based on the verification results of whether the predicted characteristic parameters meet the physical laws; obtaining the boundary condition error loss based on the verification results of whether the predicted characteristic parameters meet the boundary condition samples; determining the orthogonality constraint loss based on the verification results of whether different predicted characteristic parameters meet the orthogonality conditions; and obtaining the comprehensive loss value based on the weighted sum of the characteristic parameter error loss, the physical information error loss, the boundary condition error loss, and the orthogonality constraint loss.

[0220] In some embodiments, the comprehensive loss value includes an eigenvalue error loss; wherein, the converging the deep learning network based on the comprehensive loss value to obtain the task processing model includes: determining the eigenvalue error loss based on the difference between the predicted eigenvalue and the true eigenvalue; performing eigenvalue search according to the eigenvalue error loss and a preset eigenvalue error index through a reinforcement learning method to obtain an adjustment step and adjustment direction for the next eigenvalue search; obtaining a predicted eigenvalue corresponding to the next eigenvalue search based on the adjustment step and adjustment direction of the next eigenvalue search, and the predicted eigenvalue; continuing to train the deep learning network based on the predicted eigenvalue corresponding to the next eigenvalue search, and returning to the step of determining the eigenvalue error loss based on the difference between the predicted eigenvalue and the true eigenvalue, and determining the comprehensive loss value based on the eigenvalue error loss, converging the deep learning network based on the comprehensive loss value to obtain the task processing model.

[0221] In some embodiments, the eigenvalue search is performed according to the eigenvalue error loss and the preset eigenvalue error index by the reinforcement learning method to obtain the adjustment step size for the next eigenvalue search, including: when the current search is the first search stage, the adjustment step size for the next eigenvalue search is determined to be the first adjustment step size according to the eigenvalue error loss and the preset eigenvalue error index by the reinforcement learning method; when the current search is the second search stage, the adjustment step size for the next eigenvalue search is determined to be the second adjustment step size according to the eigenvalue error loss and the eigenvalue error index by the reinforcement learning method; wherein, the first adjustment step size is larger than the second adjustment step size, and the second search stage is located after the first search stage.

[0222] In some embodiments, the predicted characteristic parameters include a predicted characteristic function, and the training step further includes: selecting multiple monitoring points within the calculation area of ​​the Sturm-Liouville equation, and determining the characteristic function error loss corresponding to the predicted characteristic function at each monitoring point, the calculation area is determined according to the boundary condition sample; according to the characteristic function error loss at each monitoring point, the error control strategy in the corresponding calculation area is adjusted.

[0223] In some embodiments, the modeling parameters include at least one of boundary conditions, weight functions, potential functions and operator parameters, and the characteristic parameters include eigenvalues ​​and eigenfunctions; when the target task includes the elastic modal analysis task, the boundary conditions include that the displacement at both ends of the elastic object is zero; the weight function includes the density distribution of the elastic object; the potential function includes the equivalent potential energy distribution of the tension of the elastic object and the external restoring force; the operator parameters include the ratio of the mass per unit length to the tension of the elastic object; the eigenvalue includes the square of the vibration frequency; the eigenfunction includes the vibration mode shape of the elastic object at the vibration frequency; the physical laws include at least one of energy conservation, momentum balance, material homogeneity assumption, system linear small vibration assumption, solution continuity and solution differentiability; when the target task includes the heat conduction analysis task, the boundary conditions include that the boundary of the object maintains a constant temperature or an adiabatic state; the weight function includes the heat capacity distribution of the object material; the potential function includes the spatial variation of thermal conductivity; the operator parameters include at least one of thermal conductivity, density and specific heat capacity; the eigenvalue includes thermal diffusivity; the The characteristic function includes the temperature distribution pattern in the stable state; the physical law includes at least one of Fourier's law of heat conduction, energy conservation, material property constraints, continuity of solution and differentiability of solution; when the target task includes the electromagnetic wave propagation characteristic analysis task, the boundary condition includes the interface condition of the propagation medium; the weight function includes magnetic permeability; the potential function includes dielectric constant; the operator parameter includes the square of the wave number in vacuum; the eigenvalue includes propagation constant; the characteristic function includes the distribution of magnetic field in the medium; the physical law includes Maxwell's law that electromagnetic waves must satisfy At least one of a system of equations, boundary conditions of a medium, continuity of a solution, and differentiability of a solution; when the target task includes the particle state distribution task, the boundary conditions are determined according to the application scenario; the weight function includes mass density distribution; the potential function includes the potential energy distribution of the environment in which the particle is located; the operator parameters include Planck's constant and / or particle mass; the eigenvalue includes the energy eigenvalue of the particle; the eigenfunction includes the wave function form under the energy eigenvalue; the physical law includes at least one of the Schrödinger equation, the probability conservation principle, continuity of a solution, and differentiability of a solution.

[0224] The task processing device provided in this embodiment is based on the same concept as the task processing method provided in the above embodiments of this application. It can execute the task processing method provided in any of the above embodiments of this application and has the corresponding functional modules and beneficial effects of executing the task processing method. For technical details not fully described in this embodiment, please refer to the specific processing content of the task processing method provided in the above embodiments of this application, and will not be repeated here.

[0225] The functions implemented by the above task processing unit 701 can be implemented by the same or different processors respectively, and the embodiment of the present application is not limited thereto.

[0226] It should be understood that the units in the above devices can be implemented in the form of a processor calling software. For example, the device includes a processor, the processor is connected to a memory, and the memory stores instructions. The processor calls the instructions stored in the memory to implement any of the above methods or realize the functions of each unit of the device. The processor can be a general-purpose processor, such as a CPU or a microprocessor, and the memory can be a memory within the device or a memory outside the device. Alternatively, the units in the device can be implemented in the form of hardware circuits. The functions of some or all units can be realized by designing the hardware circuits. The hardware circuit can be understood as one or more processors. For example, in one implementation, the hardware circuit is an ASIC, and the functions of some or all of the above units can be realized by designing the logical relationships between the components within the circuit. For another example, in another implementation, the hardware circuit can be implemented by a PLD. For example, an FPGA can include a large number of logic gate circuits. The connection relationships between the logic gate circuits are configured through a configuration file to realize the functions of some or all of the above units. All units of the above devices can be implemented entirely in the form of a processor calling software, or entirely in the form of hardware circuits, or partially in the form of a processor calling software, with the remaining parts implemented in the form of hardware circuits.

[0227] In an embodiment of the present application, a processor is a circuit with the ability to process signals. In one implementation, the processor may be a circuit with the ability to read and execute instructions, such as a CPU, a microprocessor, a GPU, or a DSP. In another implementation, the processor may implement certain functions through the logical relationship of a hardware circuit, and the logical relationship of the hardware circuit may be fixed or reconfigurable, such as a hardware circuit implemented by an ASIC or PLD, such as an FPGA. In a reconfigurable hardware circuit, the process of the processor loading a configuration document to implement the configuration of the hardware circuit can be understood as the process of the processor loading instructions to implement the functions of some or all of the above units. In addition, it can also be a hardware circuit designed for artificial intelligence, which can be understood as an ASIC, such as an NPU, TPU, DPU, etc.

[0228] It can be seen that each unit in the above device can be one or more processors (or processing circuits) configured to implement the above method, such as: CPU, GPU, NPU, TPU, DPU, microprocessor, DSP, ASIC, FPGA, or a combination of at least two of these processor forms.

[0229] In addition, the various units in the above apparatus may be fully or partially integrated together, or may be implemented independently. In one implementation, these units are integrated together and implemented in the form of a system-on-chip (SOC). The SOC may include at least one processor for implementing any of the above methods or implementing the functions of the various units of the apparatus. The at least one processor may be of different types, such as a CPU and an FPGA, a CPU and an artificial intelligence processor, a CPU and a GPU, etc.

[0230] Exemplary electronic devices

[0231] The present application embodiment provides an electronic device, see Figure 8 As shown, the electronic device includes: a memory 200 and a processor 210; wherein the memory 200 is connected to the processor 210 and is used to store programs; the processor 210 is used to implement the task processing method disclosed in any of the above embodiments by running the program stored in the memory 200.

[0232] Specifically, the electronic device may further include: a bus, a communication interface 220 , an input device 230 and an output device 240 .

[0233] The processor 210, the memory 200, the communication interface 220, the input device 230 and the output device 240 are interconnected via a bus. The bus may include a path for transmitting information between various components of the computer system.

[0234] Processor 210 can be a general-purpose processor, such as a general-purpose central processing unit (CPU), a microprocessor, or the like. It can also be an application-specific integrated circuit (ASIC), or one or more integrated circuits used to control the execution of the program of the present invention. It can also be a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic device, discrete gate or transistor logic device, or discrete hardware components.

[0235] The processor 210 may include a main processor, and may also include a baseband chip, a modem, and the like.

[0236] Memory 200 stores programs that implement the technical solutions of the present invention and may also store an operating system and other key services. Specifically, the programs may include program code, which includes computer operating instructions. More specifically, memory 200 may include read-only memory (ROM), other types of static storage devices capable of storing static information and instructions, random access memory (RAM), other types of dynamic storage devices capable of storing information and instructions, disk storage, flash memory, and the like.

[0237] The input device 230 may include a device for receiving data and information input by a user, such as a keyboard, a mouse, a camera, a scanner, a light pen, a voice input device, a touch screen, a pedometer, or a gravity sensor.

[0238] Output device 240 may include devices that allow information to be output to a user, such as a display screen, printer, speakers, etc.

[0239] The communication interface 220 may include any device such as a transceiver to communicate with other devices or communication networks, such as Ethernet, a radio access network (RAN), a wireless local area network (WLAN), etc.

[0240] The processor 210 executes the program stored in the memory 200 and calls other devices, and can be used to implement each step of any task processing method provided in the above embodiments of the present application.

[0241] An embodiment of the present application also proposes a chip, which includes a processor and a data interface. The processor reads and runs a program stored in a memory through the data interface to execute the task processing method introduced in any of the above embodiments. The specific processing process and its beneficial effects can be found in the embodiment introduction of the above-mentioned task processing method.

[0242] Exemplary computer program products and storage media

[0243] In addition to the above-mentioned methods and devices, an embodiment of the present application may also be a computer program product, which includes computer program instructions, which, when executed by a processor, enable the processor to execute the steps of the task processing method according to various embodiments of the present application described in any of the above-mentioned embodiments of this specification.

[0244] The computer program product may be written in any combination of one or more programming languages ​​to implement the program code for performing the operations of the embodiments of the present application, including object-oriented programming languages ​​such as Java, C++, and conventional procedural programming languages ​​such as "C" or similar programming languages. The program code may be executed entirely on the user's computing device, partially on the user's computing device, as a standalone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server.

[0245] In addition, an embodiment of the present application may also be a storage medium on which a computer program is stored. The computer program is executed by a processor to execute the steps of the task processing method according to various embodiments of the present application described in any of the above embodiments of this specification, and specifically can implement the steps of the embodiments of the above task processing method.

[0246] For the sake of simplicity, the aforementioned method embodiments are described as a series of action combinations. However, those skilled in the art should be aware that this application is not limited by the order of the actions described, because according to this application, certain steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in this specification are all preferred embodiments, and the actions and modules involved are not necessarily required by this application.

[0247] It should be noted that the various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. Similarities between the various embodiments can be referred to in conjunction with each other. For device embodiments, since they are generally similar to method embodiments, their description is relatively simple, and relevant details can be referred to the description of the method embodiments.

[0248] The steps in the methods of the various embodiments of the present application can be adjusted in order, combined, and deleted according to actual needs, and the technical features recorded in the various embodiments can be replaced or combined. The modules and submodules in the devices and terminals of the various embodiments of the present application can be combined, divided, and deleted according to actual needs.

[0249] In the several embodiments provided in this application, it should be understood that the disclosed terminals, devices, and methods can be implemented in other ways. For example, the terminal embodiments described above are merely illustrative. For example, the division of modules or submodules is merely a logical function division. In actual implementation, there may be other division methods, such as multiple submodules or modules can be combined or integrated into another module, or some features can be ignored or not executed. In addition, the mutual coupling or direct coupling or communication connection shown or discussed can be an indirect coupling or communication connection through some interface, device or module, which can be electrical, mechanical or other forms.

[0250] Modules or submodules described as separate components may or may not be physically separate, and components described as modules or submodules may or may not be physical modules or submodules, that is, they may be located in one place or distributed across multiple network modules or submodules. Some or all of the modules or submodules may be selected to achieve the objectives of this embodiment as needed.

[0251] In addition, each functional module or submodule in each embodiment of the present application may be integrated into a processing module, or each module or submodule may exist physically separately, or two or more modules or submodules may be integrated into a single module. The above-mentioned integrated modules or submodules may be implemented in the form of hardware or software functional modules or submodules.

[0252] Professionals may further appreciate that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, computer software, or a combination of the two. In order to clearly illustrate the interchangeability of hardware and software, the above description has generally described the components and steps of each example according to their functions. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professionals and technicians may use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0253] The steps of the methods or algorithms described in conjunction with the embodiments disclosed herein may be implemented directly using hardware, software executed by a processor, or a combination of the two. The software may be stored in random access memory (RAM), internal memory, read-only memory (ROM), electrically programmable ROM, electrically erasable programmable ROM, registers, a hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art.

[0254] Finally, it should be noted that, in this document, relational terms such as first and second, etc., are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprises," "comprising," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or device comprising a series of elements includes not only those elements, but also other elements not explicitly listed, or elements inherent to such process, method, article, or device. In the absence of further limitations, an element defined by the phrase "comprising a ..." does not exclude the presence of additional identical elements in the process, method, article, or device comprising the element.

[0255] The above description of the disclosed embodiments will enable those skilled in the art to implement or use the present application. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, the present application is not limited to the embodiments shown herein, but is to be construed in the widest manner consistent with the principles and novel features disclosed herein.

Claims

1. A task processing method, characterized in that: include: Obtaining modeling parameters required for modeling a target task using the Sturm-Liouville equation, wherein the target task includes at least one of an elastic modal analysis task, a heat conduction analysis task, an electromagnetic wave propagation characteristic analysis task, and a particle state analysis task; Solving the characteristic parameters of the Sturm-Liouville equation according to the modeling parameters and the physical laws corresponding to the target task through a task processing model to obtain a task processing result, wherein the characteristic parameters represent the solution target corresponding to the target task, and the solution target satisfies the physical laws; the task processing model includes an autoencoder, a physical information neural network, and a Transformer network, and the characteristic parameters include eigenvalues ​​and characteristic functions; The step of solving the characteristic parameters of the Sturm-Liouville equation according to the modeling parameters and the physical laws corresponding to the target task using the task processing model to obtain the task processing result includes: Generating, by the autoencoder according to the modeling parameters, initial eigenvalues ​​and initial eigenfunctions corresponding to the modeling parameters; Adjusting the initial characteristic function according to the physical law through the physical information neural network to obtain an intermediate characteristic function; Long-distance modeling is performed on the intermediate feature function through the Transformer network to obtain the feature function, and the initial feature value is used as the feature value.

2. The method according to claim 1, characterized in that The autoencoder includes an encoder and a decoder; The step of generating the initial eigenvalues ​​and initial eigenfunctions corresponding to the modeling parameters according to the modeling parameters by the autoencoder includes: Mapping the modeling parameters to a low-dimensional latent space through the encoder to obtain latent variables in the latent space; The initial eigenvalue and the initial eigenfunction are generated by the decoder according to latent variables in the latent space.

3. The method according to claim 1, characterized in that The task processing model is trained using the following steps: Solving the Sturm-Liouville equation using a deep learning network based on modeling parameter samples to obtain predicted characteristic parameters; the modeling parameter samples correspond to real characteristic parameters, and the modeling parameter samples include boundary condition samples; Obtaining a comprehensive loss value corresponding to a loss function based on a difference between the predicted characteristic parameters and the actual characteristic parameters, a verification result of the predicted characteristic parameters based on the boundary condition samples, and a verification result of the predicted characteristic parameters based on the physical laws; The deep learning network is converged based on the comprehensive loss value to obtain the task processing model.

4. The method according to claim 3, characterized in that The obtaining of a comprehensive loss value corresponding to a loss function based on the difference between the predicted characteristic parameters and the actual characteristic parameters, the verification result of the predicted characteristic parameters based on the boundary condition samples, and the verification result of the predicted characteristic parameters based on the physical laws includes: determining a feature parameter error loss based on a difference between the predicted feature parameter and the true feature parameter; Obtaining a physical information error loss based on a verification result of whether the predicted characteristic parameters satisfy the physical law; Obtaining a boundary condition error loss based on a verification result of whether the predicted characteristic parameter satisfies the boundary condition sample; Determining an orthogonality constraint loss based on verification results of whether different prediction feature parameters satisfy an orthogonality condition; The comprehensive loss value is obtained based on the weighted sum of the characteristic parameter error loss, the physical information error loss, the boundary condition error loss and the orthogonality constraint loss.

5. The method according to claim 3, characterized in that The comprehensive loss value includes the eigenvalue error loss, the predicted characteristic parameter includes the predicted characteristic value, and the real characteristic parameter includes the real characteristic value; The step of converging the deep learning network based on the comprehensive loss value to obtain the task processing model includes: determining the eigenvalue error loss based on a difference between the predicted eigenvalue and the true eigenvalue; Performing eigenvalue search based on the eigenvalue error loss and a preset eigenvalue error index by a reinforcement learning method to obtain an adjustment step size and an adjustment direction for the next eigenvalue search; Obtaining a predicted eigenvalue corresponding to the next eigenvalue search according to the adjustment step size and adjustment direction of the next eigenvalue search and the predicted eigenvalue; Continue training the deep learning network based on the predicted eigenvalue corresponding to the next eigenvalue search, return to the step of determining the eigenvalue error loss based on the difference between the predicted eigenvalue and the true eigenvalue, and determine the comprehensive loss value based on the eigenvalue error loss, converge the deep learning network based on the comprehensive loss value, and obtain the task processing model.

6. The method according to claim 5, characterized in that The step of performing eigenvalue search according to the eigenvalue error loss and a preset eigenvalue error index by the reinforcement learning method to obtain an adjustment step size for the next eigenvalue search includes: When the current search is in the first search stage, determining the next eigenvalue search adjustment step size as the first adjustment step size according to the eigenvalue error loss and a preset eigenvalue error index by a reinforcement learning method; When the current search is in the second search stage, determining, by a reinforcement learning method, the adjustment step size of the next eigenvalue search as the second adjustment step size according to the eigenvalue error loss and the eigenvalue error index; The first adjustment step size is greater than the second adjustment step size, and the second search phase is located after the first search phase.

7. The method according to claim 3, characterized in that The prediction feature parameter includes a prediction feature function, and the method further includes: Selecting a plurality of monitoring points within a calculation region of the Sturm-Liouville equation and determining a characteristic function error loss corresponding to a predicted characteristic function at each monitoring point, wherein the calculation region is determined according to the boundary condition sample; According to the characteristic function error loss at each monitoring point, the error control strategy in the corresponding calculation area is adjusted.

8. The method according to claim 1, characterized in that The modeling parameters include at least one of boundary conditions, weight functions, potential functions, and operator parameters; the characteristic parameters include characteristic values ​​and characteristic functions; and the method further includes: In the case where the target task includes the elastic modal analysis task, the boundary conditions include that the displacement at both ends of the elastic object is zero; the weight function includes the density distribution of the elastic object; the potential function includes the equivalent potential energy distribution of the tension of the elastic object and the external restoring force; the operator parameter includes the ratio of the mass per unit length to the tension of the elastic object; the eigenvalue includes the square of the vibration frequency; the eigenfunction includes the vibration mode shape of the elastic object at the vibration frequency; the physical law includes at least one of conservation of energy, momentum balance, material homogeneity assumption, system linear small vibration assumption, continuity of solution, and differentiability of solution; In the case where the target task includes the heat conduction analysis task, the boundary condition includes that the boundary of the object maintains a constant temperature or an adiabatic state; the weight function includes the heat capacity distribution of the object material; the potential function includes the spatial variation of thermal conductivity; the operator parameter includes at least one of thermal conductivity, density, and specific heat capacity; the eigenvalue includes thermal diffusivity; the eigenfunction includes a temperature distribution pattern in a steady state; and the physical law includes at least one of Fourier's heat conduction law, energy conservation, material property constraints, continuity of solution, and differentiability of solution; In the case where the target task includes the electromagnetic wave propagation characteristics analysis task, the boundary conditions include interface conditions of the propagation medium; the weight function includes magnetic permeability; the potential function includes dielectric constant; the operator parameter includes the square of the wave number in a vacuum; the eigenvalue includes the propagation constant; the eigenfunction includes the distribution of the magnetic field in the medium; and the physical law includes at least one of the Maxwell equations that the electromagnetic wave must satisfy, the boundary conditions of the medium, the continuity of the solution, and the differentiability of the solution; When the target task includes the particle state distribution task, the boundary conditions are determined according to the application scenario; the weight function includes the mass density distribution; the potential function includes the potential energy distribution of the environment in which the particle is located; the operator parameters include the Planck constant and / or the particle mass; the eigenvalue includes the energy eigenvalue of the particle; the eigenfunction includes the wave function form under the energy eigenvalue; the physical law includes at least one of the Schrödinger equation, the probability conservation principle, the continuity of the solution and the differentiability of the solution.

9. A task processing device, characterized in that: include: A task processing unit, configured to execute the method according to any one of claims 1 to 8.

10. An electronic device, characterized in that: including memory and processor; The memory is connected to the processor and is used to store programs; The processor is configured to implement the method according to any one of claims 1 to 8 by running the program in the memory.

11. A computer program product, characterized in that The method comprises computer program instructions, which, when executed by a processor, cause the processor to implement the method according to any one of claims 1 to 8.

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