A mathematical modeling method and system based on an artificial intelligence large model
By combining an automated mathematical modeling engine with Math-LLM, the problems of low efficiency and poor model versatility in traditional modeling are solved, achieving efficient and interpretable mathematical modeling that is suitable for complex industrial systems and supports online updates and integration.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- TONGJI UNIV
- Filing Date
- 2026-02-06
- Publication Date
- 2026-06-02
AI Technical Summary
Traditional mathematical modeling relies on domain experts, which is inefficient and cannot quickly respond to the modeling needs of complex systems. The models have poor generality and transferability, making it difficult to handle high-dimensional, nonlinear, and heterogeneous data. Furthermore, existing data-driven models lack interpretability and physical constraints, making them difficult to integrate into industrial applications.
An automated mathematical modeling engine is built, which collects and preprocesses terminal data, calls the Math-LLM large mathematical model to generate candidate models that conform to mathematical and physical logic, performs double verification and optimal selection, and combines data-driven and physics-driven fusion modeling methods to achieve model automation and knowledge transfer by using reinforcement learning and pre-trained datasets.
It achieves efficient and interpretable mathematical modeling, reduces reliance on expert experience, improves modeling efficiency and the robustness and versatility of models, supports online learning and industrial application integration, and can handle complex systems and output structured models.
Smart Images
Figure CN122133459A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of artificial intelligence technology, and in particular relates to a mathematical modeling method and system based on a large artificial intelligence model. Background Technology
[0002] In recent years, mathematical modeling has become a core component of scientific research and industrial engineering (such as process optimization, equipment failure prediction, and financial market analysis). Mathematical modeling is the process of describing the behavior of a real-world system using mathematical language (such as equations, inequalities, and logical relationships), serving as a bridge between the physical and digital worlds. In industry, mathematical models are widely used for simulation, prediction, control, and optimization; for example, building temperature distribution models for steelmaking furnaces to optimize energy consumption and creating degradation models for wind power equipment for predictive maintenance.
[0003] Traditional modeling paradigms heavily rely on "white-box" or "grey-box" modeling methods employed by domain experts. The manual modeling process by domain experts can be broken down into: problem understanding -> data collection and cleaning -> hypothesis formulation and equation (model structure) design -> parameter estimation -> model validation and debugging. Traditional modeling processes have the following inherent flaws: Highly dependent on experts and inefficient: Because the formulation of the model structure relies heavily on experts' prior knowledge and personal experience of physical / chemical / biological mechanisms, it results in long modeling cycles, high human resource costs, and an inability to respond quickly to the modeling needs of different scenarios on a large scale. Insufficient ability to handle complex systems: For complex industrial systems with unclear mechanisms, strong nonlinear coupling between variables, or high-dimensional and heterogeneous data (such as chemical reaction processes and health management of large equipment groups), experts find it difficult to construct accurate mathematical models based on experience. This often leads to existing technologies using oversimplified models, sacrificing accuracy and generalization ability. Poor model versatility and transferability: Because each model is "tailor-made" for a specific problem, its knowledge (i.e., modeling logic and equation form) cannot be accumulated and reused. When encountering a similar but slightly different new problem, experts need to repeat the entire modeling process almost from scratch, resulting in a waste of knowledge. Model maintenance and updates are difficult: When system equipment is upgraded, process conditions change, or new data is generated, the original manually built model often becomes invalid, requiring experts to intervene again for tedious adjustments or reconstruction, making it impossible to achieve self-evolution and online adaptation of the model.
[0004] Purely data-driven "black box" machine learning models (such as neural networks and random forests) are directly used as predictive models due to their powerful fitting capabilities. However, these models lack interpretability, are difficult to embed into simulation or optimization systems requiring explicit physical constraints, and their predictive behavior may violate fundamental physical laws, limiting their application in critical industrial scenarios. They suffer from the following inherent drawbacks: Its modeling depth is limited: its core remains "curve fitting" rather than "model discovery." It selects a predefined model structure (framework) and then adjusts the parameters. Unlike human experts, it cannot actively "invent" new, reasonable differential equations, partial differential equations, or complex structures such as state-space models with latent variables from first principles or data. Insufficient mathematical rigor: Symbolic expressions obtained through searches such as genetic programming, while having a certain degree of readability, often lack mathematical rigor (such as inconsistent dimensions and violation of basic conservation laws), and the search process is extremely computationally expensive, easily getting trapped in local optima and producing complex expressions that have no physical meaning. Lack of a unified knowledge base: Each modeling task is independent, and the system lacks the ability to learn and transfer knowledge from a large, structured mathematical knowledge base (such as existing physical laws and common equation forms). Each modeling task starts from scratch. Low integration with industrial scenarios: The output model is usually just a prediction function, which is difficult to directly convert into a standard mathematical model format that can be used in process simulation software (such as ASPEN, COMSOL) or control systems. Integration and application require secondary development.
[0005] Given the shortcomings of the existing technologies, there is an urgent need to develop a mathematical modeling method and system based on large-scale artificial intelligence models. Summary of the Invention
[0006] The purpose of this invention is to provide a mathematical modeling method and system based on a large artificial intelligence model to solve the technical problems in the prior art.
[0007] To address the aforementioned technical problems, this invention provides a mathematical modeling method based on a large-scale artificial intelligence model, comprising the following steps: S1. Construct an automated mathematical modeling engine for end-to-end data connection with users in terminal application scenarios; S2. The automated mathematical modeling engine collects application scenario data and questions input by end users; S3. The automated mathematical modeling engine performs data preprocessing and feature extraction on the application scenario data. S4. The automated mathematical modeling engine calls the Math-LLM large mathematical model to generate multiple candidate models that conform to mathematical and physical logic; S5. The automated mathematical modeling engine performs both mathematical and physical verification and optimal selection on the candidate models; S6. Output the final interpretable mathematical model, fitted parameter values, model performance, and hypothesis report to the end user.
[0008] Preferably, step S3 specifically comprises: S31. Data cleaning: handling missing values and outliers; S32. Automatic Feature Engineering: Generate derived features based on time series and spectrum, including mean, variance, and frequency band energy; S33. Feature Dimensionality Reduction and Selection: Use PCA and mutual information to select the feature set that has the greatest predictive power for the target variable; S34. Output a clean, high-quality dataset of feature variables as input to the mathematical model Math-LLM.
[0009] Preferably, step S4 specifically comprises: S41. Using prompt word engineering, the application scenario problem, feature variable dataset, and domain physical constraints are constructed into precise instructions for input to the mathematical large model Math-LLM; S42. The mathematical model Math-LLM is subjected to chain thinking to generate multiple candidate model structures that are mathematically and physically reasonable, including different equation forms.
[0010] Preferably, step S5 specifically comprises: S51. Model Validation: This includes numerical error screening or probabilistic screening. Numerical error screening fits the candidate model to the user input data and calculates quantitative indicators, including fitting error and generalization error. Probabilistic screening introduces Bayesian model averaging or a weighted average method based on information criteria to output a probabilistic combination of multiple models. S52. Symbolic simplification: Using a computer algebra system to perform equivalent transformations, combine like terms, and simplify expressions on equations generated by the large mathematical model Math-LLM. S53. Physical consistency check: Verify the dimensional homogeneity of the final model, the rationality of the boundary conditions, and whether it strictly follows the physical constraints; S54. Model selection: Select the optimal model by comprehensively considering numerical accuracy, simplicity and physical consistency.
[0011] Preferably, the large mathematical model Math-LLM possesses powerful mathematical capabilities, and the training steps are as follows: Pre-training: Using mathematical modeling datasets for pre-training; Supervised fine-tuning: Supervised fine-tuning is performed on input specific user task data, which includes private data submitted by users and manually labeled data from organizations; Reinforcement learning: Reinforcement learning is performed on input user task data.
[0012] Preferably, the steps for constructing the mathematical modeling dataset are as follows: Collect raw data, which comes from mathematical text and code corpora and industrial simulation and experimental datasets; The raw data is parsed and extracted, including OCR text parsing and time-series data segmentation; Semi-automatic annotation and alignment are performed, including initial algorithmic matching, manual verification / correction, and data augmentation. Output standardized JSON format data.
[0013] Preferably, the Math-LLM architecture of the large mathematical model adopts a standard Decoder-only Transformer, Encoder-Decoder architecture, or a neural symbolic hybrid system.
[0014] Preferably, the automated mathematical modeling engine can call the large mathematical model Math-LLM generated using a large language model, and can also call the graph neural network GNN, treating variables as nodes in a graph and unknown relationships between variables as edges. By learning the types of edges through GNN, including linear, product, differential and weight, a computational graph representing the equation can be directly generated.
[0015] Another technical solution provided by this invention: a mathematical modeling system based on a large artificial intelligence model, the system adopting a layered architecture, comprising, in sequence: The modeling dataset module is used to build mathematical modeling datasets; The large mathematical model module is used to train the large mathematical model Math-LLM based on the mathematical modeling dataset and to perform inference modeling based on the input content; The automated mathematical modeling engine module is used to coordinate the work of the entire system, transform application scenario problems into tasks that the large mathematical model module can understand, and manage the entire process of model generation, verification and output. The terminal application module is used to realize information interaction between the automated mathematical modeling engine module and the end user.
[0016] Preferably, the automated mathematical modeling engine module includes: The issue module is used to collect application scenario data and issues input by end users; The data preprocessing and feature extraction module is used to perform data preprocessing and feature extraction on the application scenario data. The large model generation and inference module is used to call the Math-LLM mathematical large model to generate multiple candidate models that conform to mathematical and physical logic; The model validation and simplification module is used to perform both mathematical and physical validation on the candidate models and to select the optimal one.
[0017] Compared with the prior art, the beneficial effects of the present invention are: This invention directs the symbol generation capabilities of large language models to the field of serious industrial application mathematical modeling, overcoming the problem of lack of interpretability in traditional AI models (such as deep learning black boxes).
[0018] An automated, interactive "human-machine collaborative modeling" pipeline has been established. Engineers only need to provide the problem and data, and the system can, like a skilled mathematical modeling expert, propose and verify multiple hypotheses, greatly reducing the professional threshold and improving modeling efficiency.
[0019] It achieves a fusion of data-driven and physics-driven approaches. The final model is both faithful to the data patterns and rooted in physical laws, thus possessing greater robustness and extrapolation capabilities.
[0020] This invention transforms the mathematical modeling process, which originally relied on the experience of senior experts, into a standardized, repeatable, and scalable automated service, providing a solid "model foundation" for industrial intelligence.
[0021] By addressing the issues of reliance on experts and efficiency, and by constructing a "mathematical big model" with powerful mathematical reasoning and modeling capabilities, the system can automatically or semi-automatically derive potential mathematical relationships and model structures from data. This significantly reduces or even eliminates the reliance on the modeling experience of domain experts, shortening modeling time from the "month / year" level to the "hour / day" level, and significantly improving modeling efficiency.
[0022] This invention enhances the ability to handle complex problems and utilizes AI technologies such as deep learning to process high-dimensional, nonlinear, and heterogeneous data. It can automatically discover complex patterns and hidden variable relationships that are difficult to capture by traditional methods, thereby building more accurate and reliable mathematical models for complex industrial systems.
[0023] This approach enables the accumulation and transfer of model knowledge. By constructing a unified, high-quality mathematical dataset for pre-training the "mathematical large model," the model learns general mathematical symbol operations, equation transformations, and modeling paradigms. When faced with new problems, this large model can quickly transfer learned knowledge, achieving "train once, apply in multiple places" and improving the versatility of modeling.
[0024] The system achieves automated model maintenance and supports online and incremental learning. When new data is introduced, the system can automatically evaluate the performance of the existing model and trigger fine-tuning or reconstruction processes, enabling the mathematical model to dynamically adapt to changes in the environment and system, and achieve continuous optimization.
[0025] The modeling depth and automation level of this invention are fundamentally different: existing technologies (such as AutoML) select and tune parameters from a preset family of models, which is a "model selection" process; while this invention uses Math-LLM to actively "reason" and "invent" new model structures from data, which is closer to the creative modeling process of human experts, has a higher degree of automation, and can handle more complex modeling tasks.
[0026] The model of this invention offers superior interpretability and physical consistency: existing symbolic regression methods may yield meaningless expressions. This invention embeds mathematical and physical common sense into Math-LLM through training, resulting in candidate equations that are mathematically more reasonable. Furthermore, after screening and verification by the physical constraint module, the final model exhibits stronger interpretability and physical consistency, making it more readily accepted by domain experts.
[0027] This invention possesses knowledge transfer and continuous learning capabilities: existing models operate independently for each task, while the Math-LLM of this invention serves as a unified knowledge carrier. Through pre-training on a large number of cross-domain mathematical modeling tasks, it masters general modeling skills and can quickly transfer them when facing new domain problems, demonstrating the ability to "learn by analogy". Furthermore, the system supports online fine-tuning, enabling continuous evolution of the model.
[0028] This invention integrates more closely with industrial processes: It not only outputs a prediction function, but also a structured mathematical model, which can be automatically converted into an industrial software-compatible format. It can be seamlessly integrated into existing simulation, optimization and control workflows, significantly enhancing its practicality. Attached Figure Description
[0029] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.
[0030] Figure 1 A flowchart illustrating the steps of a mathematical modeling method based on a large artificial intelligence model, provided for embodiments of this application; Figure 2 A flowchart illustrating the steps for constructing a mathematical modeling dataset provided in this application embodiment; Figure 3 A flowchart illustrating the construction and training steps of a large mathematical model provided in this application embodiment; Figure 4 A schematic diagram of the module structure of a mathematical modeling system based on a large artificial intelligence model, provided for an embodiment of this application; Figure 5A schematic diagram of the module structure of the automated mathematical modeling engine module provided in the embodiments of this application; Figure 6 This is a schematic diagram illustrating the operation of the terminal application module provided in an embodiment of this application. Detailed Implementation
[0031] In the description of this application, "multiple" means at least two, such as two, three, etc., unless otherwise explicitly specified. All directional indications (such as up, down, left, right, front, back, top, bottom, etc.) in the embodiments of this application are only used to explain the relative positional relationships and movement of the components in a specific posture (as shown in the figures). If the specific posture changes, the directional indication will also change accordingly. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion. For example, a process, method, system, product, or IoT terminal that includes a series of steps or units is not limited to the listed steps or units, but may optionally include steps or units not listed, or may optionally include other steps or units inherent to these processes, methods, products, or IoT terminals.
[0032] Furthermore, the reference to "embodiment" herein means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.
[0033] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0034] Example 1 like Figure 1 As shown, this invention provides a mathematical modeling method based on a large-scale artificial intelligence model, comprising the following steps: S1. Construct an automated mathematical modeling engine that connects to end-to-end user data in terminal application scenarios. This automated mathematical modeling engine is key to engineering and streamlining Math-LLM capabilities.
[0035] S2. The automated mathematical modeling engine collects application scenario data and questions input by end users. Users submit raw industrial data (such as sensor time-series data) and describe the problem in natural language (similar to the natural language description in the large mathematical model module), such as predicting the remaining lifespan of the spindle.
[0036] S3. The automated mathematical modeling engine performs data preprocessing and feature extraction on the application scenario data.
[0037] S4. The automated mathematical modeling engine calls the Math-LLM large mathematical model to generate multiple candidate models that conform to mathematical and physical logic.
[0038] S5. The automated mathematical modeling engine performs both mathematical and physical verification and selects the optimal model.
[0039] S6. Output the final interpretable mathematical model (such as a concise differential equation), fitted parameter values, and a report on model performance and assumptions to the end user.
[0040] This method can achieve a closed loop in the core technical process of "structuring industrial problems -> data characterization -> generating mathematical models using Math-LLM -> dual verification by mathematics and physics".
[0041] As a preferred embodiment of the present invention, step S3 specifically comprises: S31. Data cleaning: handling missing values and outliers; S32. Automatic Feature Engineering: Generate derived features based on time series, spectrum, etc., such as mean, variance, and frequency band energy; S33. Feature Dimensionality Reduction and Selection: Use PCA, mutual information, and other methods to select the feature set that has the greatest predictive power for the target variable; S34. Output a clean, high-quality dataset of feature variables as input to the mathematical model Math-LLM.
[0042] As a preferred embodiment of the present invention, step S4 specifically comprises: S41. Prompt Engineering: Construct the application scenario problem, feature variable dataset, and domain physical constraints into precise instructions (Prompts) input to the Math-LLM mathematical model. For example: "Given the feature variables {X1: vibration energy, X2: temperature gradient, ...}, the data trend shows exponential decay, and the system needs to satisfy the constraint of non-negative energy decay. Please derive an ordinary differential equation describing its evolution." S42. Multi-round reasoning and generation: The mathematical big model Math-LLM performs chain thinking based on prompts to generate multiple mathematically and physically reasonable candidate model structures, including different equation forms.
[0043] As a preferred embodiment of the present invention, step S5 specifically comprises: S51. Model Validation: This includes numerical error screening or probabilistic screening. In numerical error screening, the candidate model is fitted to the user input data, and quantitative indicators are calculated, including fitting error and generalization error. Probabilistic screening introduces Bayesian model averaging (BMA) or a weighted averaging method based on information criteria (such as AIC, BIC), outputting a probabilistic combination of multiple models. Probabilistic screening is used to quantify model uncertainty, not to determine a single "optimal" model.
[0044] S52. Symbolic simplification: Using a computer algebra system (CAS) to perform equivalent transformations, combine like terms, and simplify expressions on equations generated by the large mathematical model Math-LLM; S53. Physical consistency check: Verify the dimensional homogeneity of the final model, the rationality of the boundary conditions, and whether it strictly follows the physical constraints; S54. Model selection: Select the optimal model by comprehensively considering numerical accuracy, simplicity (Occam's razor), and physical consistency.
[0045] As a preferred technical solution of the present invention, the mathematical large-scale model Math-LLM is the core intelligent agent, not the general ChatGPT, but a "domain expert" specializing in mathematical ability, possessing powerful mathematical capabilities. It is trained by calling data from the modeling dataset module to obtain a mathematical large-scale model with powerful mathematical capabilities, whose core capabilities include: Natural Language Understanding: Parsing problem descriptions about application scenarios such as industrial processes.
[0046] Generate mathematical models: Based on the user's input question, the large mathematical model generates a structured mathematical model (equations, inequalities, etc.).
[0047] The large mathematical model is trained through three processes: pre-training, supervised fine-tuning, and reinforcement learning. Pre-training data comes from mathematical text and code corpora, and industrial simulation and experimental datasets, which can be obtained through the internet or relevant data companies. Supervised fine-tuning data comes from data specific to a user task; it can use user-submitted private data or manually labeled data. Reinforcement learning data also comes from data specific to a user task; it can use user-submitted private data or manually labeled data. Figure 3 The diagram illustrates the three-stage training process for a large mathematical model. The detailed training steps are as follows: Pre-training: The mathematical modeling dataset is used for pre-training. This dataset is responsible for building high-quality vertical domain datasets, which is a prerequisite for Math-LLM to have industrial modeling capabilities.
[0048] Supervised fine-tuning: Supervised fine-tuning is performed on input specific user task data, which includes private data submitted by users and manually labeled data from organizations.
[0049] Reinforcement learning: Reinforcement learning is performed on input user task data.
[0050] Users can submit private data, and each piece of data in this section is a "task sample" in the following format: Input: `{Natural language description of the problem, relevant industrial data snippets, list of known physical constraints}` Output: `{candidate mathematical model (e.g., system of differential equations), parameter range, model assumptions}` As a preferred technical solution of the present invention, such as Figure 2 The diagram illustrates the processing flow of key data resources used for training Math-LLM, from their raw form to a standardized task dataset. The steps for constructing the mathematical modeling dataset are as follows: Raw data was collected from mathematical text and code corpora and industrial simulation and experimental datasets. The raw data formats included mathematical literature PDFs and historical industrial data. The mathematical text and code corpora contained textbooks, papers, and open-source code (such as SymPy and MATLAB scripts) used for pre-training Math-LLM, enabling it to master basic mathematical language, symbolic computation rules, and equation derivation logic. The industrial simulation and experimental datasets covered time-series data, parameter tables, and images from multiple operating conditions and devices, providing a realistic distribution of industrial data. This data could be obtained through the internet or from relevant data companies and was used for pre-training large-scale models.
[0051] The raw data is parsed and extracted, including using OCR text parsing and time-series data segmentation.
[0052] Semi-automatic labeling and alignment can be achieved through initial matching using algorithms, manual verification / correction, and data augmentation.
[0053] Output standardized JSON format data. Standardized JSON format data typically looks like this: "id": "task_001" "context": "Describes a spring-damped system..." "variables":["x", "t", "m", "c", "k"] “data”:[[t0, x0], ...] / / Optional simulated or real data "model":"m*d^2x / dt^2 + c*dx / dt + k*x = 0" As a preferred technical solution of the present invention, the Math-LLM architecture is not limited to the standard Decoder-only Transformer (such as the GPT series), but can also adopt an Encoder-Decoder architecture (such as T5), or even a neural symbolic hybrid system, for example, a neural network is responsible for extracting features and relationships from the data, and a symbolic engine is responsible for assembling equations according to rules, and the two work together.
[0054] As a preferred technical solution of the present invention, the automated mathematical modeling engine can not only call the large mathematical model Math-LLM generated using a large language model, but also call the modeling scheme generated using a graph neural network (GNN). It treats variables as nodes in a graph and unknown relationships between variables as edges. It learns the types of edges through GNN, including linear, product, differential, and weight, thereby directly generating a computational graph representing the equation.
[0055] Example 2 like Figure 4 As shown, this invention provides a mathematical modeling system based on a large-scale artificial intelligence model. The system adopts a classic layered architecture, realizing the value transfer from data and models to applications, and includes the following components: The modeling dataset module is used to build mathematical modeling datasets. It is the system's "fuel library" and provides training data and task definitions for upper-level models.
[0056] The Mathematical Large Model Module (Math-LLM) is used to train the Math-LLM mathematical large model based on the mathematical modeling dataset and to perform reasoning modeling based on the input content. It is the "intelligent brain" of the system and has professional mathematical modeling and symbolic reasoning capabilities.
[0057] The automated mathematical modeling engine module is used to coordinate the work of the entire system. It is the "coordination center" and "execution arm" of the system, transforming application scenario problems into tasks that the large mathematical model module can understand, and managing the entire process of model generation, verification and output.
[0058] The terminal application module connects and enables information interaction between the automated mathematical modeling engine and the end user. It serves as the system's "value output," providing mathematical models to the user and solving specific problems. Its mechanism involves calling the services of the "automated mathematical modeling engine" through standardized API interfaces to address specific application scenarios, such as industrial problems.
[0059] like Figure 6As shown, the mathematical modeling engine is used by terminal application modules via API calls to solve specific application scenario problems. In addition to providing APIs, the system can be integrated as a plugin into commonly used scientific computing environments (such as Jupyter Notebook, MATLAB) or industrial IoT platforms, allowing users to directly call automated modeling functions in familiar environments.
[0060] Typical applications: Predictive maintenance: Input historical operating data of the equipment to automatically generate a "health degradation equation" to predict the time of failure.
[0061] Process optimization: Input production parameters and product quality data, and automatically discover the implicit functional relationship between "quality and process parameters" to guide parameter tuning.
[0062] Simulation and Design: Rapidly build hybrid models that integrate first-principles and data for new equipment or processes, accelerating virtual testing.
[0063] like Figure 5 As shown, the automated mathematical modeling engine module includes: The Problem module is used to collect application scenario data and problems input by end users. Users input problems through a web interface or API, upload data tables, and describe the problems in natural language. Users submit raw industrial data (such as sensor time-series data) and describe the problems in natural language (similar to the natural language description in the mathematical model module), such as predicting the remaining life of a spindle.
[0064] The data preprocessing and feature extraction module is used to preprocess and extract features from the application scenario data. The system preprocesses and extracts features from the user-input data, including data cleaning, automatic feature engineering, feature dimensionality reduction and selection, and finally outputs a clean, high-quality feature variable dataset as the "input" to the large model generation and inference module.
[0065] The large model generation and inference module is used to call the Math-LLM mathematical model to generate multiple candidate models that conform to mathematical and physical logic. Calling the large mathematical model generation and inference module allows for the dynamic construction of prompts for the large mathematical model, which then generates and infers based on these prompts.
[0066] The model validation and simplification module is used to perform both mathematical and physical validation on the candidate models and to select the optimal one. The final model output is selected through numerical fitting and validation, symbol simplification, and physical checks.
[0067] Descriptions that are repeated in Example 1 are omitted here.
[0068] The core of this system lies in: building a high-quality mathematical dataset -> training a large artificial intelligence model (Math-LLM) with powerful mathematical capabilities -> building an end-to-end automated mathematical modeling engine for application scenarios such as industry.
[0069] Example 3 The following specific example will be used to illustrate the above technical solutions and processes.
[0070] Assume the industrial scenario is: "Based on the historical vibration signals, inlet pressure, outlet pressure, and flow data of a centrifugal pump, establish a predictive model for its health status (degradation index)."
[0071] Step S1: Input question: Users upload a table of sensor timing data for a centrifugal pump via a web interface or API, and describe the question in natural language: "Establish a model of the relationship between vibration amplitude and operating conditions (pressure, flow rate) and equipment health status to predict degradation."
[0072] Step S2: Data preprocessing and feature extraction process: 1. Data cleaning: handling missing values and outliers.
[0073] 2. Automatic feature engineering: Generate derived features (such as mean, variance, and frequency band energy) based on time series and spectrum.
[0074] 3. Feature dimensionality reduction and selection: Use PCA, mutual information method, etc. to screen out the feature set with the most predictive power for the target variable.
[0075] Output: A clean, high-quality dataset of feature variables, which serves as the "input" to the large model generation and inference modules.
[0076] Step S3: Large model generation and inference (core) process: The Math-LLM (Mathematical Large Model) is invoked. This Math-LLM is a Transformer architecture model trained on high-quality mathematical modeling task datasets. The model generation and inference are achieved through the following process: 1. Prompt word engineering: Based on the output of step S2, dynamically construct prompt words for Math-LLM. For example: "You are a mathematical modeling expert. Given the following variables: eigenvector X (containing vibration frequency domain features F1, F2, pressure P, and flow rate Q), and target variable: degradation index D (range 0-1). Data shows that D monotonically increases with time, and vibration energy increases with increasing D. Based on ordinary differential equations (ODEs) or algebraic equations, please propose three possible mathematical models to describe the dynamic evolution relationship between D and X. Please ensure that the equations are mathematically sound." 2. Large Model Inference: Prompt words are input into Math-LLM. Math-LLM utilizes its knowledge learned from the modeling dataset module to perform "thought chain" reasoning, generating multiple candidate equations. For example: Candidate 1: `dD / dt = a * F1 * (1 - D)` (damage accumulation model) Candidate 2: `D = 1 - exp(-b * ∫(F2 * P) dt)` (Degeneracy model based on cumulative energy) Candidate 3: `D = c0 + c1*F1 + c2*Q^2 + c3*sin(ω*t)` (Empirical model with periodic terms) Step S4: Model validation and simplification process: 1. Numerical Fitting and Validation: For each candidate equation, using historical input data, the parameters (a, b, c0, c1...) in the equation are estimated by nonlinear least squares or Bayesian inference. Then, the prediction error (e.g., RMSE) and stability index are calculated on the retained validation dataset.
[0077] 2. Symbolic simplification and physical checks: Symbolic computation is used to simplify the fitted equations. For example, checking whether the dimensions of all equations are consistent.
[0078] 3. Model Selection and Output: The optimal model is selected by comprehensively evaluating goodness of fit, simplicity (Occam's Razor), and physical consistency. The final output includes an "interpretable mathematical equation," "fit parameter values," a "model confidence report," and a "model file in a standard format" (such as a Functional Mock-up Unit, FMU, or Python function). For example, the final output equation is: `dD / dt=0.005*(high-frequency vibration energy - 4.2)^2 * (1-D)`, with the physical meaning that "when the high-frequency vibration energy exceeds the threshold of 4.2, the equipment degradation rate is proportional to the square of the excess energy, and the degradation rate slows down as D increases."
[0079] Through the above steps, the system has completed the automated construction of an interpretable and deployable mathematical model from raw industrial data.
[0080] This application employs a method for constructing high-quality mathematical modeling datasets and a method for generating specific format datasets for training Math-LLM. This includes a technical solution for constructing a paired dataset of `(problem context, mathematical equation / model structure)` from publicly available mathematical literature, simulation software output, and historical industrial data through semi-automatic annotation (such as aligning equations in text with corresponding descriptions) and data augmentation (such as adding noise or perturbation to known equations to generate new samples).
[0081] This application employs a training architecture and method for a large model with mathematical reasoning capabilities (Math-LLM) and a Transformer model training method specifically designed for mathematical modeling, including but not limited to: using mathematical symbols as special tokens for embedding; introducing "symbolic logic loss" during the model training phase (requiring the model to predict the next legal mathematical symbol operation); and using "contrastive learning" to allow the model to distinguish between physically reasonable and unreasonable equation structures.
[0082] This application employs an automated modeling engine workflow that integrates domain constraints, particularly the specific technical implementation of transforming domain knowledge (such as physical constraints) into machine-understandable, embeddable prompts for the model.
[0083] This application adopts a hybrid modeling approach that combines large model inference with numerical verification. Specifically, it describes how the system uses Math-LLM to generate candidate model structures, then combines traditional numerical optimization methods for parameter estimation and verification, and iteratively optimizes the model structure through feedback loops (such as re-inputting the verification error as part of the prompt word into the LLM).
[0084] This application adopts a model output and packaging format for industrial applications, which automatically converts the final mathematical model into an industry-standard simulation or deployment format, realizing the automation of the "last mile" from AI model to engineering-usable mathematical model.
[0085] The fundamental difference between the large mathematical model in this application and general-purpose large language models (such as ChatGPT) is that, although the underlying technologies are similar, this invention emphasizes "specialization": Specialized training data: Paired data is modeled using highly structured mathematical models, rather than generic web page text.
[0086] Capability Objective Specific: The objective is to generate correct, concise, and physically interpretable mathematical expressions, rather than fluent open-domain text.
[0087] Evaluation metrics are specific to this study: the evaluation criteria are the mathematical correctness, fitting accuracy, and physical consistency of the generative model, rather than the fluency of the text.
[0088] The hierarchical interpretation of "automation" in this application: In this invention, "automation" can be of different levels: Fully automated: Users only need to provide data and a simple problem description, and the system will automatically run the entire process.
[0089] Human-computer interaction: At key nodes (such as after candidate model generation), the system provides multiple options and explains their physical meaning, allowing experts to select or modify them before continuing the subsequent process. This interaction mode should also fall within the scope of protection of this invention.
[0090] This invention provides a systematic solution centered on a large mathematical model, integrating domain knowledge, and oriented towards end-to-end application scenarios. This invention directs the symbol generation capabilities of a large language model towards the serious field of mathematical modeling in industrial applications, overcoming the lack of interpretability in traditional AI models (such as the black box of deep learning).
[0091] An automated, interactive "human-machine collaborative modeling" pipeline has been established. Engineers only need to provide the problem and data, and the system can, like a skilled mathematical modeling expert, propose and verify multiple hypotheses, greatly reducing the professional threshold and improving modeling efficiency.
[0092] It achieves a fusion of data-driven and physics-driven approaches. The final model is both faithful to the data patterns and rooted in physical laws, thus possessing greater robustness and extrapolation capabilities.
[0093] This invention transforms the mathematical modeling process, which originally relied on the experience of senior experts, into a standardized, repeatable, and scalable automated service, providing a solid "model foundation" for industrial intelligence.
[0094] The basic principles of this application have been described above with reference to specific embodiments. However, it should be noted that the advantages, benefits, and effects mentioned in this application are merely examples and not limitations, and should not be considered as essential features of each embodiment of this application. Furthermore, the specific details disclosed above are for illustrative and facilitative purposes only, and are not limitations. These details do not limit the application to the necessity of employing the specific details described above. The above description is provided to enable any person skilled in the art to make or use this application. Various modifications to these aspects will be readily apparent to those skilled in the art, and the general principles defined herein can be applied to other aspects without departing from the scope of this application. Therefore, this application is not intended to be limited to the aspects shown herein, but rather to be accorded the widest scope consistent with the principles and novel features of this application.
[0095] The above are merely preferred embodiments of this application and are not intended to limit the scope of this application. Any modifications or equivalent substitutions made within the spirit and principles of this application shall be included within the protection scope of this application.
[0096] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.
Claims
1. A mathematical modeling method based on a large-scale artificial intelligence model, characterized in that, Including the following steps: S1. Construct an automated mathematical modeling engine for end-to-end data connection with users in terminal application scenarios; S2. The automated mathematical modeling engine collects application scenario data and questions input by end users; S3. The automated mathematical modeling engine performs data preprocessing and feature extraction on the application scenario data. S4. The automated mathematical modeling engine calls the Math-LLM large mathematical model to generate multiple candidate models that conform to mathematical and physical logic; S5. The automated mathematical modeling engine performs both mathematical and physical verification and optimal selection on the candidate models; S6. Output the final interpretable mathematical model, fitted parameter values, model performance, and hypothesis report to the end user.
2. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 1, characterized in that, Step S3 specifically involves: S31. Data cleaning: handling missing values and outliers; S32. Automatic Feature Engineering: Generate derived features based on time series and spectrum, including mean, variance, and frequency band energy; S33. Feature Dimensionality Reduction and Selection: Use PCA and mutual information to select the feature set that has the greatest predictive power for the target variable; S34. Output a clean, high-quality dataset of feature variables as input to the mathematical model Math-LLM.
3. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 2, characterized in that, Step S4 specifically involves: S41. Using prompt word engineering, the application scenario problem, feature variable dataset, and domain physical constraints are constructed into precise instructions for input to the mathematical large model Math-LLM; S42. The mathematical model Math-LLM is subjected to chain thinking to generate multiple candidate model structures that are mathematically and physically reasonable, including different equation forms.
4. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 1, characterized in that, Step S5 specifically involves: S51. Model Validation: This includes numerical error screening or probability screening. Numerical error screening fits the candidate model to the user input data and calculates quantitative indicators, including fitting error and generalization error. Probabilistic screening introduces Bayesian model averaging or information criterion-based weighted averaging methods to output a probabilistic combination of multiple models. S52. Symbolic simplification: Using a computer algebra system to perform equivalent transformations, combine like terms, and simplify expressions on equations generated by the large mathematical model Math-LLM. S53. Physical consistency check: Verify the dimensional homogeneity of the final model, the rationality of the boundary conditions, and whether it strictly follows the physical constraints; S54. Model selection: Select the optimal model by comprehensively considering numerical accuracy, simplicity and physical consistency.
5. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 1, characterized in that, The Math-LLM mathematical model described above possesses powerful mathematical capabilities. The training steps are as follows: Pre-training: Using mathematical modeling datasets for pre-training; Supervised fine-tuning: Supervised fine-tuning is performed on input specific user task data, which includes private data submitted by users and manually labeled data from organizations; Reinforcement learning: Reinforcement learning is performed on input user task data.
6. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 5, characterized in that, The steps for constructing the mathematical modeling dataset are as follows: Collect raw data, which comes from mathematical text and code corpora and industrial simulation and experimental datasets; The raw data is parsed and extracted, including OCR text parsing and time-series data segmentation; Semi-automatic annotation and alignment are performed, including initial algorithmic matching, manual verification / correction, and data augmentation. Output standardized JSON format data.
7. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 1, characterized in that, The Math-LLM architecture, a large mathematical model, employs standard Decoder-only Transformer, Encoder-Decoder architecture, or a hybrid neural symbolic system.
8. The mathematical modeling method based on a large-scale artificial intelligence model according to claim 1, characterized in that, The automated mathematical modeling engine can call the large mathematical model Math-LLM generated using a large language model, and it can also call the graph neural network GNN, treating variables as nodes in a graph and unknown relationships between variables as edges. By learning the types of edges through GNN, including linear, product, differential, and weight, a computational graph representing the equation can be directly generated.
9. A mathematical modeling system based on a large-scale artificial intelligence model, characterized in that, The system adopts a layered architecture, comprising, in sequence: The modeling dataset module is used to build mathematical modeling datasets; The large mathematical model module is used to train the large mathematical model Math-LLM based on the mathematical modeling dataset and to perform inference modeling based on the input content; The automated mathematical modeling engine module is used to coordinate the work of the entire system, transform application scenario problems into tasks that the large mathematical model module can understand, and manage the entire process of model generation, verification and output. The terminal application module is used to realize information interaction between the automated mathematical modeling engine module and the end user.
10. A mathematical modeling system based on a large-scale artificial intelligence model according to claim 9, characterized in that, The automated mathematical modeling engine module includes: The issue module is used to collect application scenario data and issues input by end users; The data preprocessing and feature extraction module is used to perform data preprocessing and feature extraction on the application scenario data. The large model generation and inference module is used to call the Math-LLM mathematical large model to generate multiple candidate models that conform to mathematical and physical logic; The model validation and simplification module is used to perform both mathematical and physical validation on the candidate models and to select the optimal one.