A method and apparatus for constructing material constitutive equations based on large language models

Abstract

This application provides a method for constructing material constitutive equations based on a large language model, which can be applied to the field of artificial intelligence technology. The method includes: acquiring a stress-strain experimental dataset of materials; performing data analysis on the experimental dataset and a current set of equations based on a first large language model to generate a natural language strategy; inputting the natural language strategy into a second large language model to output a set of candidate equations; performing multi-objective Pareto optimization evaluation on the set of candidate equations, considering fitting error, generalization error, and equation complexity, and updating the Pareto optimal set of equations through non-dominated sorting; feeding the updated Pareto optimal set of equations back to the first large language model, and repeatedly executing the strategy generation and candidate equation generation operations until a preset convergence condition is met; and outputting the Pareto optimal set of equations as the Pareto front of the material constitutive equations. This application also provides a device for constructing material constitutive equations based on a large language model.

Description

Technical Field

[0001] This application relates to the field of artificial intelligence, specifically to the field of symbolic regression technology, and more specifically to a method, apparatus, device, medium, and program product for constructing material constitutive equations based on a large language model. Background Technology

[0002] Symbolic Regression (SR) is a class of algorithms that automatically discover latent mathematical equations from data and is widely used in scientific modeling fields such as physics, chemistry, biology, economics, and materials science. Its core objective is to take observational data as input and output an explicit mathematical expression that can accurately predict and is interpretable.

[0003] Traditional methods rely on human experience or symbolic regression based on genetic programming, which suffers from low search efficiency, susceptibility to local optima, and difficulty in controlling model complexity. With the development of artificial intelligence, symbolic regression methods based on large language models have emerged, utilizing the contextual learning capabilities and pre-trained knowledge of these models to generate equations. However, these methods lack a systematic data prior extraction mechanism, failing to effectively leverage the statistical characteristics and causal dependencies of stress-strain data to guide the search direction; furthermore, their optimization objective is singular, focusing solely on fitting error, leading to equations prone to overfitting and poor interpretability. Therefore, a novel method is urgently needed that can automatically analyze data patterns, embed physical priors, and continuously evolve the equation structure through closed-loop optimization. Summary of the Invention

[0004] In view of the above problems, this application provides a method, apparatus, device, medium and program product for constructing material constitutive equations based on large language models.

[0005] According to the first aspect of this application, a method for constructing material constitutive equations based on a large language model is provided, including:

[0006] The process involves: acquiring a stress-strain experimental dataset of the material; performing data analysis on the experimental dataset and the current set of equations based on a first large language model to generate a natural language strategy. The first large language model is used to extract variable coupling relationships representing the dependencies between variables and structural pattern features representing the regularity of the mathematical structure of the current set of equations; inputting the natural language strategy into a second large language model to output a set of candidate equations; performing multi-objective Pareto optimization evaluation on the set of candidate equations, considering fitting error, generalization error, and equation complexity, and updating the Pareto optimal set of equations through non-dominated sorting; feeding the updated Pareto optimal set of equations back to the first large language model, and repeating the strategy generation and candidate equation generation operations until a preset convergence condition is met; and outputting the Pareto optimal set of equations as the Pareto front of the material constitutive equations, where the Pareto front provides a set of candidate equation solutions that exhibit a non-dominated relationship between fitting error, generalization ability, and equation complexity.

[0007] According to an embodiment of this application, the multi-objective Pareto optimization evaluation of the candidate equation set, including fitting error, generalization error, and equation complexity, includes: independently calculating the fitting error index, generalization error index, and equation complexity index for each equation in the candidate equation set; constructing the fitting error index, generalization error index, and equation complexity index together as a three-dimensional optimization objective; and performing solution set optimization using a non-dominated ranking strategy, wherein the ranking strategy determines whether any candidate equation is not greater than other equations in all three indices and whether the value of at least one index is less than the value of the corresponding index of other equations.

[0008] According to an embodiment of this application, the determination process of the non-dominated ranking strategy includes: randomly selecting two candidate equations and comparing them item by item with three indicators; if one equation is not greater than the other equation in terms of fitting error, generalization error and complexity indicators, and the value of at least one indicator is less than the value of the corresponding indicator of the other equation, then the former is determined to dominate the latter; after traversing all candidate equations, the equations that do not have any dominance relationship are retained in the Pareto optimal set.

[0009] According to an embodiment of this application, updating the Pareto optimal equation set by non-dominated sorting includes: merging the candidate equation set and the historical Pareto optimal equations into a temporary solution set; performing non-dominated sorting on the temporary solution set to remove dominated equations and update the current Pareto front.

[0010] According to an embodiment of this application, the step of performing data analysis on the experimental dataset and the current set of equations based on a first large language model to generate a natural language strategy includes: calling a data analysis tool to extract variable coupling relationships from the experimental dataset to characterize the dependencies between variables; performing structural pattern analysis on the current set of equations to obtain structural pattern features; and generating a natural language strategy based on the variable coupling relationships and the structural pattern features.

[0011] According to embodiments of this application, the data analysis tools include a linear correlation analysis tool, a nonlinear correlation analysis tool, a time-frequency analysis tool, and a causal inference tool. The linear correlation analysis tool is used to extract the Pearson correlation coefficient between stress and strain; the nonlinear correlation analysis tool is used to extract the Spearman rank correlation coefficient between variables; the time-frequency analysis tool is used to identify the frequency components and periodicity patterns in stress-strain data; and the causal inference tool is used to determine the causal dependence of the strain history on the stress response.

[0012] A second aspect of this application provides a constitutive equation construction apparatus for materials based on a large language model, comprising:

[0013] The system comprises the following modules: an acquisition module for acquiring experimental datasets of material stress and strain; a natural language strategy generation module for performing data analysis on the experimental dataset and the current set of equations based on a first large language model to generate natural language strategies, wherein the first large language model is used to extract variable coupling relationships representing the dependencies between variables and structural pattern features representing the regularity of the mathematical structure of the current set of equations; a first output module for inputting the natural language strategies into a second large language model to output a set of candidate equations; a multi-objective evaluation module for performing multi-objective Pareto optimization evaluation on the set of candidate equations for fitting error, generalization error, and equation complexity, and updating the Pareto optimal set of equations through non-dominated sorting; an iterative update module for feeding back the updated Pareto optimal set of equations to the first large language model, and repeatedly executing strategy generation and candidate equation generation operations until a preset convergence condition is reached; and a second output module for outputting the Pareto front of the material constitutive equations formed by the Pareto optimal set of equations, wherein the Pareto front is used to provide a set of candidate equation solutions that exhibit a non-dominated relationship between fitting error, generalization ability, and equation complexity.

[0014] According to embodiments of this application, the multi-objective evaluation module includes a calculation submodule and an optimization submodule.

[0015] The calculation submodule is used to independently calculate the fitting error index, generalization error index, and equation complexity index for each equation in the candidate equation set; the optimization submodule is used to construct the fitting error index, generalization error index, and equation complexity index into a three-dimensional optimization objective, and to perform solution set optimization using a non-dominated sorting strategy. The sorting strategy determines whether any candidate equation is not greater than other equations in all three indices and whether the value of at least one index is less than the value of the corresponding index of other equations.

[0016] According to an embodiment of this application, the optimization submodule is further used to randomly select two candidate equations and compare them item by item using three indicators. If one of the equations is no greater than the other equation in terms of fitting error, generalization error and complexity indicators, and the value of at least one indicator is less than the value of the corresponding indicator of the other, then it is determined that the former dominates the latter. After traversing all candidate equations, the equations that do not have any dominance relationship are retained in the Pareto optimal set.

[0017] According to an embodiment of this application, the optimization submodule is further configured to merge the candidate equation set and the historical Pareto optimal equations into a temporary solution set; and to perform a non-dominated sort on the temporary solution set to remove dominated equations and update the current Pareto front.

[0018] According to embodiments of this application, the natural language policy generation module includes a data analysis submodule, a structural pattern analysis submodule, and a generation submodule.

[0019] The data analysis submodule is used to call data analysis tools to extract variable coupling relationships from the experimental dataset to characterize the dependencies between variables; the structural pattern analysis submodule is used to perform structural pattern analysis on the current set of equations to obtain structural pattern features; and the generation submodule is used to generate natural language strategies based on the variable coupling relationships and the structural pattern features.

[0020] A third aspect of this application provides an electronic device comprising: one or more processors; and a memory for storing one or more computer programs, wherein the one or more processors execute the one or more computer programs to implement the steps of the method described above.

[0021] A fourth aspect of this application also provides a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implement the steps of the above-described method.

[0022] The fifth aspect of this application also provides a computer program product, including a computer program or instructions that, when executed by a processor, implement the steps of the above-described method. Attached Figure Description

[0023] The above-mentioned contents, other objects, features and advantages of this application will become clearer from the following description of embodiments with reference to the accompanying drawings, in which:

[0024] Figure 1 A flowchart illustrating a method for constructing material constitutive equations based on a large language model according to an embodiment of this application is shown schematically.

[0025] Figure 2 This illustration shows a flowchart of a method for generating a natural language strategy based on a first large language model to perform data analysis on the experimental dataset and the current set of equations, according to an embodiment of this application.

[0026] Figure 3 A flowchart illustrating a multi-objective Pareto optimization evaluation method according to an embodiment of this application is shown schematically.

[0027] Figure 4 A schematic diagram illustrates the structural block diagram of a material constitutive equation construction device based on a large language model according to an embodiment of this application; and

[0028] Figure 5 A block diagram schematically illustrates an electronic device suitable for implementing a method for constructing material constitutive equations based on a large language model, according to an embodiment of this application. Detailed Implementation

[0029] The embodiments of this application will now be described with reference to the accompanying drawings. However, it should be understood that these descriptions are exemplary only and are not intended to limit the scope of this application. In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the embodiments of this application for ease of explanation. However, it will be apparent that one or more embodiments may be implemented without these specific details. Furthermore, descriptions of well-known structures and technologies are omitted in the following description to avoid unnecessarily obscuring the concepts of this application.

[0030] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of this application. The terms “comprising,” “including,” etc., as used herein indicate the presence of the stated features, steps, operations, and / or components, but do not exclude the presence or addition of one or more other features, steps, operations, or components.

[0031] All terms used herein (including technical and scientific terms) have the meanings commonly understood by those skilled in the art, unless otherwise defined. It should be noted that the terms used herein are to be interpreted in a manner consistent with the context of this specification, and not in an idealized or overly rigid way.

[0032] When using expressions such as "at least one of A, B and C", they should generally be interpreted in accordance with the meaning that is commonly understood by those skilled in the art (e.g., "a system having at least one of A, B and C" should include, but is not limited to, a system having A alone, a system having B alone, a system having C alone, a system having A and B, a system having A and C, a system having B and C, and / or a system having A, B and C, etc.).

[0033] The constitutive equation of a material is the core mathematical model describing the stress-strain relationship, directly determining the accuracy of finite element simulation, structural safety assessment, and life prediction. Traditional methods rely on human experience or symbolic regression based on genetic programming, which suffers from low search efficiency, susceptibility to local optima, and difficulty in controlling model complexity. In recent years, although large language models have shown potential in symbolic regression, related technologies still face three bottlenecks: First, the lack of a systematic data prior extraction mechanism makes it impossible to effectively utilize the statistical characteristics and causal dependencies of stress-strain data to guide the search direction; second, the optimization objective is singular, focusing only on fitting error, leading to overfitting and poor interpretability of the generated equations; third, the single-generation mode lacks iterative update capabilities, making it difficult to achieve a dynamic trade-off between accuracy, generalization, and complexity. Especially in modeling materials such as the plasticity of metals and the viscoelasticity of polymers, experimental costs are high and physical constraints are strict.

[0034] In view of the above-mentioned technical problems, embodiments of this application provide a method for constructing material constitutive equations based on a large language model, which involves: acquiring a stress-strain experimental dataset of materials; performing data analysis on the experimental dataset and the current set of equations based on a first large language model to generate a natural language strategy, wherein the first large language model is used to extract variable coupling relationships representing the dependencies between variables and structural pattern features representing the regularity of the mathematical structure of the current set of equations; inputting the natural language strategy into a second large language model to output a set of candidate equations; performing multi-objective Pareto optimization evaluation on the set of candidate equations based on fitting error, generalization error, and equation complexity, and updating the Pareto optimal set of equations through non-dominated sorting; feeding the updated Pareto optimal set of equations back to the first large language model, and repeating the strategy generation and candidate equation generation operations until a preset convergence condition is reached; and outputting the Pareto optimal set of equations as the Pareto front of the material constitutive equations, wherein the Pareto front is used to provide a set of candidate equation solutions that exhibit a non-dominated relationship between fitting error, generalization ability, and equation complexity.

[0035] Figure 1 A flowchart illustrating a method for constructing material constitutive equations based on a large language model according to an embodiment of this application is shown.

[0036] like Figure 1 As shown, the method of this embodiment includes operations S110 to S160.

[0037] In operation S110, obtain the stress-strain experimental dataset of the material.

[0038] In one example, a uniaxial tensile test of titanium alloy at 600-800℃ and a strain rate of 0.001-1s⁻¹ is used to illustrate the specific implementation process. An isothermal constant strain rate compression test was conducted on a cylindrical titanium alloy specimen (Φ8 mm × 12 mm) using a thermal simulation testing machine, and stress-strain data were collected. The raw data were then smoothed and filtered to construct a dataset.

[0039] , where the strain vector Includes strain, strain rate, and temperature. The corresponding rheological stress is given. The dataset is divided into a training set Dtrain and a validation set Dval in an 8:2 ratio.

[0040] In operation S120, data analysis is performed on the experimental dataset and the current set of equations based on the first large language model to generate a natural language strategy.

[0041] According to an embodiment of this application, the first large language model is used to extract variable coupling relationships that characterize the dependencies between variables and structural pattern features that characterize the regularity of the current set of equations in mathematical structure.

[0042] In one example, the first major language model, LLM1, performs two analysis tasks on the experimental dataset and the current set of equations: variable coupling extraction and structural pattern feature analysis. Specifically, LLM1 automatically invokes data analysis tools to analyze the strength of linear dependencies, non-monotonic coupling relationships, periodicity features, and causal analysis between variables. These analysis results are then converted into prior knowledge in natural language, such as a strong positive correlation between stress and strain hardening (Pearson coefficient 0.92), significant softening at temperatures above 700°C (Spearman coefficient -0.78), a periodic component in the strain rate of 1.2 Hz, and the causal relationship between the strain rate of the previous moment and the current stress response.

[0043] In one example, structural pattern feature analysis is performed on the current set of equations, which is the current Pareto optimal set of equations P. t When initially an empty set, it can be a baseline equation randomly generated from a large model, statistically analyzing high-frequency node types, if P tIf more than 30% of the equations contain the exp(−ε) structure, the generation mode description is as follows: The current optimal equation set generally contains exponential decay terms. It is recommended to strengthen the exploration of this term in the generation of new equations and try its composite form with polynomial terms. The comprehensive generation natural language strategy is: "The target constitutive equation should contain strain hardening terms, temperature softening exponent terms exp(−T / T0), and strain rate coupling terms ε˙." m Prioritize exploring the multiplicative combination structure of exponential terms and polynomials.

[0044] In operation S130, the natural language strategy is input into the second large language model to output a set of candidate equations.

[0045] In operation S140, the candidate equation set is evaluated by multi-objective Pareto optimization based on fitting error, generalization error, and equation complexity, and the Pareto optimal equation set is updated by non-dominated sorting.

[0046] In operation S150, the updated Pareto optimal equation set is fed back to the first language model, and the policy generation and candidate equation generation operations are repeated until the preset convergence condition is reached.

[0047] In operation S160, the Pareto optimal equation set is output to form the Pareto front of the material constitutive equation. The Pareto front is used to provide a set of candidate equation solutions that have a non-dominant relationship between fitting error, generalization ability and equation complexity.

[0048] In one example, the policy generated by operation S120 is used as input to the second largest language model, LLM2, and the output is a set of candidate equations. For each candidate equation Calculate the three-dimensional evaluation index:

[0049] 1) Intra-domain fitting error (NMSE_ID)

[0050]

[0051] in, Represent the candidate symbolic equation with respect to the input. The predicted value; Represents the actual observed value; Represents the in-domain region of the training dataset; Indicates the output of the training set. The mean of the training set is the value of the training set fitting error. The smaller the value, the lower the fitting error of the training set.

[0052] 2) Out-of-domain generalization error (NMSE_OOD)

[0053] definition:

[0054] in, This represents the Out-of-Domain region in the dataset. It is constructed as follows: for each input dimension... Calculate quantiles (e.g., the 5th percentile and the 95th percentile); classify samples falling outside this interval as the OOD region, and the remaining samples as the ID region. NMSE_OOD measures the extrapolation ability of an equation, that is, the accuracy of prediction under unseen data distributions; scientific laws are often universal, so an excellent equation should maintain low error in the OOD region, and the smaller the value, the stronger the extrapolation ability of the equation.

[0055] 3) Equation Complexity (ASTLen)

[0056] in, Equation Abstract Syntax Tree (AST); This represents the set of nodes in a tree. ASTLen is the total number of nodes. Fewer nodes indicate a simpler equation structure.

[0057] In one example, each equation in the candidate equation set is sequentially evaluated using the three metrics mentioned above. After calculating the three metrics, a non-dominated ranking determination is performed. Specifically, two equations are randomly selected from the candidate equation set and compared item by item across the three metrics. If one equation is no greater than the other equation in terms of fitting error, generalization error, and complexity, and at least one of its metrics is less than the corresponding metric of the other equation (the difference between the two values ​​is greater than a preset tolerance threshold), then the former is determined to dominate the latter. After traversing all equations and completing pairwise comparisons, all equations with dominance relationships are eliminated, leaving only equations without any dominance relationships to form a new Pareto optimal set. This new Pareto optimal equation set is fed back to the first language model as input, triggering a new round of policy generation. The first language model performs structural pattern analysis on the current Pareto optimal equation set, identifying frequently occurring function operator types, variable coupling methods, and coefficient distribution patterns. For example, if more than a preset proportion of equations in the current Pareto optimal set contain composite structures of exponential functions and polynomials, the recommendation strength of such structures is increased in the generated natural language strategy, while the generation probability of invalid operators verified in the previous round is reduced. Based on the updated natural language strategy, the second language model generates the next batch of candidate equation sets, and repeats the multi-objective optimization evaluation process of operation S140 until a preset convergence condition is reached. The preset convergence condition can be, for example, the number of iterations reaching a preset upper limit (e.g., 20 times), or it can be that when an equation satisfying the preset thresholds for fitting error and complexity appears in the Pareto optimal set, the convergence condition is determined to be met, and the iteration process is terminated.

[0058] In one example, after the iteration terminates, the final set of Pareto-optimal equations is output as the Pareto front of the material constitutive equations. This front contains multiple candidate equations, each of which is non-dominated in the three-dimensional evaluation index space, achieving an optimal balance in terms of fitting accuracy, generalization ability, and equation complexity. Taking the construction of the constitutive equations for the high-temperature plasticity of titanium alloys as an example, the output Pareto front may contain eight candidate equations. Some equations emphasize fitting accuracy (suitable for high-precision finite element simulation), some emphasize structural simplicity (suitable for rapid engineering estimation), and some emphasize generalization ability (suitable for extrapolation prediction). Engineers can select the equations that best suit the application scenario from the Pareto front as the final constitutive model according to specific engineering needs, and integrate it into the material model library of the finite element analysis software for numerical simulation and process parameter optimization of the hot forging process.

[0059] It should be noted that the method provided in this application is not limited to the construction of material constitutive equations. The closed-loop optimization architecture based on large language model collaboration and the multi-objective Pareto evaluation mechanism can be extended to other scientific modeling scenarios involving the discovery of complex multivariate coupling relationships. For example, in the chemical industry, it can be used to discover the correlation equations between catalyst activity and reactant concentration, temperature, and pressure; in the energy field, it can be used to construct lifetime prediction models for lithium battery capacity decay and charge / discharge rate, cycle number, and temperature; in the biomedical field, it can be used to derive the quantitative relationship between enzyme reaction kinetics and substrate concentration, pH value, and inhibitors; and in the economics field, it can be used to identify the nonlinear transmission equations between asset price fluctuations and macroeconomic indicators and market sentiment factors. This method, by replacing the experimental dataset type, adjusting the physical property extraction module of the data analysis tool, and adapting to domain constraints, can achieve automatic discovery of cross-domain symbolic regression equations, possessing broad engineering application value and scientific universality.

[0060] The closed-loop collaborative mechanism of the large language model in this application significantly shortens the material constitutive modeling cycle and reduces the amount of experimental data required. The output Pareto front provides 5-10 non-dominated candidate equations, enabling engineers to flexibly select according to the accuracy requirements of finite element simulation or the need for rapid engineering estimation, thus overcoming the deficiency of related technologies that can only output a single overfitting equation. In the plastic constitutive modeling of difficult-to-deform materials such as titanium alloys and high-temperature alloys, it reduces out-of-domain prediction errors and significantly improves model credibility and engineering applicability.

[0061] Figure 2 The flowchart illustrates a method for generating a natural language strategy based on a first large language model to perform data analysis on the experimental dataset and the current set of equations, according to an embodiment of this application.

[0062] like Figure 2 As shown, this includes operations S210 to S230.

[0063] In operation S210, a data analysis tool is invoked to extract variable coupling relationships from the experimental dataset to characterize the dependencies between variables.

[0064] According to embodiments of this application, the data analysis tool includes a linear correlation analysis tool, a nonlinear correlation analysis tool, a time-frequency analysis tool, and a causal inference tool. The linear correlation analysis tool is used to extract the Pearson correlation coefficient between stress and strain. The nonlinear correlation analysis tool is used to extract the Spearman rank correlation coefficient between variables. The time-frequency analysis tool is used to identify the frequency components and periodicity patterns in stress-strain data. The causal inference tool is used to determine the causal dependence of the strain history on the stress response.

[0065] In one example, taking the construction of the high-temperature plastic constitutive equation for titanium alloys as an example, the first major language model automatically calls four types of data analysis tools to perform in-depth mining of the experimental dataset:

[0066] Linear correlation analysis was used to calculate the Pearson correlation coefficient between true strain and true stress in the training dataset, yielding a coefficient of 0.92, indicating a strong linear positive correlation between strain hardening and flow stress. Simultaneously, the correlation coefficient between temperature and stress was calculated to be -0.85, revealing a negative linear dependence of temperature on material softening.

[0067] Nonlinear correlation analysis tools were used to perform Spearman rank correlation analysis on temperature and strain rate, revealing that the two exhibit non-monotonic coupling characteristics: at strain rates below... At high strain rates, the effect of temperature decays exponentially; however, in the high strain rate range, the temperature sensitivity tends to be constant. The tool outputs a rank correlation coefficient of 0.73 and identifies the nonlinear inflection point at the strain rate. Place.

[0068] Time-frequency analysis tools were used to perform a fast Fourier transform on the strain history data, identifying a low-frequency oscillation component of 0.8 Hz in the stress response, corresponding to the periodic softening behavior of the dynamic recrystallization process. The analysis results indicate that the dominant frequency amplitude significantly increases at temperatures exceeding 700℃, suggesting that dynamic recrystallization is more active at higher temperatures.

[0069] The causal inference tool performs Granger causality tests, determining that the strain rate history at the first two time points has a significant causal effect on the stress response (p-value < 0.01), while temperature changes have a 3-second lag effect. The tool outputs a causal dependency plot, clearly showing that strain rate is the core factor directly driving stress changes.

[0070] The above operations can systematically extract the deep coupling relationship between stress, strain, temperature and strain rate, and identify hysteresis effects and nonlinear inflection points that are difficult to detect by traditional methods.

[0071] In operation S220, structural pattern analysis is performed on the current set of equations to obtain structural pattern features.

[0072] In one example, the first large language model performs abstract syntax tree parsing on the current Pareto optimal equation set (which can be a preset set of 5 basic equations randomly generated by the large model during the initial iteration) and statistically analyzes the structural pattern features:

[0073] Function operator type analysis shows that 80% of the equations in the current optimal equation set contain the exponential function exp, 60% contain polynomial multiplication structures, and 40% contain the logarithmic function log. In particular... The presence of a temperature softening term in 80% of the equations indicates that the structure is remarkably effective.

[0074] Analysis of variable coupling methods reveals that high-performance equations commonly employ a multiplicative form to couple strain hardening and temperature softening terms, for example... The structure is stable in 70% of the equations. However, the additive coupling form only exists in equations with low complexity but poor fitting accuracy.

[0075] Analysis of the coefficient distribution pattern shows that the temperature softening coefficient The values ​​of are concentrated in the range of 650-720℃, the strain hardening exponent n is distributed in the range of 0.12-0.18, and the strain rate sensitivity coefficient m is mainly distributed between 0.1-0.15. This distribution provides a reasonable parameter search space constraint for the next generation.

[0076] Structural pattern analysis of the Pareto optimal equation set can automatically identify high-frequency efficient operators (such as the exp function, which appears in 80% of cases) and inefficient structures, enabling intelligent reduction of the search space. This mechanism improves the efficiency of candidate equation generation, reduces the proportion of invalid equations, and accelerates convergence.

[0077] In operation S230, a natural language strategy is generated based on the variable coupling relationship and the structural pattern features.

[0078] In one example, the first large language model integrates the variable coupling relationships of operation S210 with the structural pattern features of S220 to generate a natural language strategy to guide the second large language model:

[0079] The target constitutive equation should include the following core elements: strain hardening term. (n∈[0.12,0.18]) serves as the basic structure, with the temperature softening index term... As a mandatory component, the strain rate coupling term adopts a power function form. Structurally, priority should be given to exploring multiplicative coupling methods, particularly the product of strain hardening and temperature softening terms. The introduction of logarithmic functions is permitted. To handle the nonlinear behavior in the low-strain region, the coefficient of this term must be kept below 0.5 to avoid overfitting. Based on the causal analysis results, the strain rate history term {ε}(t-1) must be explicitly included. During generation, high-frequency operators exp and poly should be reused preferentially, and the use of the sine function should be restricted unless the dominant frequency matches in the time-frequency analysis. The final output should be 5-8 candidate equations that conform to the physical dimensions.

[0080] Transforming numerical analysis results into natural language strategies bridges the semantic gap between data-driven statistical analysis and symbolic equation generation. This strategy automatically updates in each iteration, improving the matching degree between candidate equations and the current optimal structure, ensuring the directionality and scientific rigor of the evolutionary process, and avoiding the blindness of traditional random search.

[0081] Figure 3 A flowchart illustrating a multi-objective Pareto optimization evaluation method according to an embodiment of this application is shown.

[0082] like Figure 3 As shown, this includes operations S310 to S340.

[0083] In operation S310, the fitting error index, generalization error index, and equation complexity index are independently calculated for each equation in the candidate equation set.

[0084] In one example, the first step is to calculate the fitting error metric. The candidate equation is substituted into each sample point in the training dataset to obtain the predicted stress value. The deviation between this predicted value and the experimentally observed stress value is calculated, and the squared deviations of all sample points are summed to obtain the total prediction error. Simultaneously, the overall variability of the experimentally observed stress values ​​is calculated as the baseline error. The total prediction error is divided by the baseline error for normalization; a smaller value indicates a better fit of the equation to the training data. The second step is to calculate the generalization error metric. For each strain rate variable, its distribution range in the training data is statistically analyzed, and data points outside this range are classified as out-of-domain test samples. Using the same calculation method as the fitting error, the prediction error is calculated on the out-of-domain test samples and normalized. This metric measures the equation's predictive ability on unseen data distributions. Finally, the equation complexity metric is calculated. The candidate equation is parsed into a tree structure representation, where each mathematical operator is a parent node and each variable or constant is a leaf node. The cumulative number of nodes in the entire tree structure is counted. Fewer nodes indicate a simpler equation structure and stronger interpretability. The fitting error index, generalization error index, and equation complexity index are calculated independently for each equation in the candidate equation set according to the above formula, which will not be elaborated here.

[0085] In operation S320, the fitting error index, generalization error index, and equation complexity index are jointly constructed into a three-dimensional optimization objective, and a non-dominated sorting strategy is used for solution set optimization.

[0086] According to an embodiment of this application, the sorting strategy determines whether any candidate equation is not greater than other equations in all three indicators and whether the value of at least one indicator is less than the value of the corresponding indicator of other equations.

[0087] According to embodiments of this application, the process for determining a non-dominant ranking strategy includes:

[0088] Two candidate equations are randomly selected and compared item by item using three indicators. If one equation is no greater than the other equation in terms of fitting error, generalization error, and complexity, and at least one indicator is less than the corresponding indicator of the other equation, then the former is determined to dominate the latter. After traversing all candidate equations, equations that do not have any dominance relationship are retained in the Pareto optimal set.

[0089] In one example, the three metrics mentioned above are constructed as a three-dimensional optimization objective vector. Since there is no single optimal solution for this objective, a non-dominated sorting strategy is used for solution set optimization.

[0090]

[0091] The determination of a non-dominant relationship follows these criteria:

[0092] Given candidate equations If the following conditions are met:

[0093]

[0094] Then it is called Dominate The term "strictly less than" here means that the difference between the two values ​​is greater than a preset tolerance threshold (such as 10 to the power of -6), thus excluding false equality judgments caused by floating-point calculation errors.

[0095] In operation S330, the candidate equation set and the historical Pareto optimal equation are merged into a temporary solution set.

[0096] In operation S340, a non-dominated sort is performed on the temporary solution set to remove dominated equations and update the current Pareto front.

[0097] In one example, the set of candidate equations generated in this round With the historical Pareto optimal equation set Merge and construct a temporary solution set This merging operation ensures that historically excellent solutions are preserved during the iteration process, preventing the loss of discovered high-performance equation structures during the evolutionary process.

[0098] For the temporary solution set U t Perform a non-dominated sort to remove all dominated equations, resulting in an updated Pareto optimal set: in This indicates the dominance relationship. The final set of output equations is the current Pareto front, and the solution set formed by it exhibits a non-dominated distribution in the three-dimensional index space, providing candidate constitutive equations that balance fitting accuracy, generalization ability, and equation complexity.

[0099] A non-dominated sorting strategy is adopted instead of the traditional weighted summation method, avoiding the bias caused by subjective weight setting. The weighted method is prone to missing the true optimal solution due to improper weight adjustment, while the non-dominated sorting strategy objectively retains all trade-off solutions, significantly increasing the proportion of true optimal solutions covered by the final Pareto front. This strategy automatically identifies and eliminates dominated redundant equations, effectively reducing the computational load per iteration and significantly accelerating the convergence speed.

[0100] By merging and sorting historical Pareto optimal equations with newly generated candidate equations, the loss of excellent solutions during the evolutionary process is effectively avoided. Traditional methods employing generational replacement strategies have a high probability of losing high-value equation structures due to random perturbations. However, the merging-screening mechanism of this invention ensures that historical optimal structural patterns always participate in strategy updates, significantly improving the effective structure inheritance rate. The final Pareto front output provides multiple non-dominated candidate equations, which engineers can flexibly select based on finite element simulation accuracy requirements or rapid engineering estimation needs, solving the pain point of existing technologies that output a single equation and cannot adapt to multiple application scenarios.

[0101] Based on the above-mentioned method for constructing material constitutive equations based on large language models, this application also provides a device for constructing material constitutive equations based on large language models. The following will be combined with... Figure 4 The device is described in detail.

[0102] Figure 4 The diagram illustrates the structure of a material constitutive equation construction apparatus based on a large language model according to an embodiment of this application.

[0103] like Figure 4 As shown, the material constitutive equation construction device 400 based on a large language model in this embodiment includes an acquisition module 410, a natural language strategy generation module 420, a first output module 430, a multi-objective evaluation module 440, an iterative update module 450, and a second output module 460.

[0104] The acquisition module 410 is used to acquire the stress-strain experimental dataset of the material. In one embodiment, the acquisition module 410 can be used to perform the operation S110 described above, which will not be repeated here.

[0105] The natural language policy generation module 420 is used to perform data analysis on the experimental dataset and the current set of equations based on a first large language model to generate a natural language policy. The first large language model is used to extract variable coupling relationships that represent the dependencies between variables and structural pattern features that represent the regularity of the mathematical structure of the current set of equations. In one embodiment, the natural language policy generation module 420 can be used to perform the operation S120 described above, which will not be repeated here.

[0106] The first output module 430 is used to input the natural language strategy into the second large language model to output a set of candidate equations. In one embodiment, the first output module 430 can be used to perform the operation S130 described above, which will not be repeated here.

[0107] The multi-objective evaluation module 440 is used to perform multi-objective Pareto optimization evaluation on the candidate equation set, considering fitting error, generalization error, and equation complexity, and to update the Pareto optimal equation set through non-dominated sorting. In one embodiment, the multi-objective evaluation module 440 can be used to perform the operation S140 described above, which will not be repeated here.

[0108] The iterative update module 450 is used to feed back the updated Pareto optimal equation set to the first large language model, and repeatedly execute the policy generation and candidate equation generation operations until the preset convergence condition is reached. In one embodiment, the iterative update module 450 can be used to execute the operation S150 described above, which will not be repeated here.

[0109] The second output module 460 is used to output the Pareto optimal equation set constituting the Pareto front of the material constitutive equation. The Pareto front is used to provide a set of candidate equation solutions that exhibit a non-dominant relationship between fitting error, generalization ability, and equation complexity. In one embodiment, the second output module 460 can be used to perform the operation S160 described above, which will not be repeated here.

[0110] According to embodiments of this application, the multi-objective evaluation module includes a calculation submodule and an optimization submodule.

[0111] The calculation submodule is used to independently calculate the fitting error index, generalization error index, and equation complexity index for each equation in the candidate equation set; the optimization submodule is used to construct the fitting error index, generalization error index, and equation complexity index into a three-dimensional optimization objective, and to perform solution set optimization using a non-dominated sorting strategy. The sorting strategy determines whether any candidate equation is not greater than other equations in all three indices and whether the value of at least one index is less than the value of the corresponding index of other equations.

[0112] According to an embodiment of this application, the optimization submodule is further used to randomly select two candidate equations and compare them item by item using three indicators. If one of the equations is no greater than the other equation in terms of fitting error, generalization error and complexity indicators, and the value of at least one indicator is less than the value of the corresponding indicator of the other, then it is determined that the former dominates the latter. After traversing all candidate equations, the equations that do not have any dominance relationship are retained in the Pareto optimal set.

[0113] According to an embodiment of this application, the optimization submodule is further configured to merge the candidate equation set and the historical Pareto optimal equations into a temporary solution set; and to perform a non-dominated sort on the temporary solution set to remove dominated equations and update the current Pareto front.

[0114] According to embodiments of this application, the natural language policy generation module includes a data analysis submodule, a structural pattern analysis submodule, and a generation submodule.

[0115] The data analysis submodule is used to call data analysis tools to extract variable coupling relationships from the experimental dataset to characterize the dependencies between variables; the structural pattern analysis submodule is used to perform structural pattern analysis on the current set of equations to obtain structural pattern features; and the generation submodule is used to generate natural language strategies based on the variable coupling relationships and the structural pattern features.

[0116] According to embodiments of this application, any multiple modules among the acquisition module 410, natural language policy generation module 420, first output module 430, multi-objective evaluation module 440, iterative update module 450, and second output module 460 can be merged into one module, or any one of these modules can be split into multiple modules. Alternatively, at least some of the functions of one or more of these modules can be combined with at least some of the functions of other modules and implemented in one module. According to embodiments of this application, at least one of the acquisition module 410, natural language policy generation module 420, first output module 430, multi-objective evaluation module 440, iterative update module 450, and second output module 460 can be at least partially implemented as hardware circuitry, such as a field-programmable gate array (FPGA), a programmable logic array (PLA), a system-on-a-chip, a system-on-a-substrate, a system-on-package, an application-specific integrated circuit (ASIC), or any other reasonable means of integrating or packaging circuitry, or implemented in software, hardware, or firmware, or in any suitable combination of any of these three implementation methods. Alternatively, at least one of the acquisition module 410, the natural language policy generation module 420, the first output module 430, the multi-objective evaluation module 440, the iterative update module 450, and the second output module 460 can be at least partially implemented as a computer program module, which can perform corresponding functions when the computer program module is run.

[0117] Figure 5 A block diagram schematically illustrates an electronic device suitable for implementing a method for constructing material constitutive equations based on a large language model, according to an embodiment of this application.

[0118] like Figure 5 As shown, an electronic device 500 according to an embodiment of this application includes a processor 501, which can perform various appropriate actions and processes according to a program stored in a read-only memory (ROM) 502 or a program loaded from a storage portion 508 into a random access memory (RAM) 503. The processor 501 may include, for example, a general-purpose microprocessor (e.g., a CPU), an instruction set processor and / or an associated chipset and / or a special-purpose microprocessor (e.g., an application-specific integrated circuit (ASIC)), etc. The processor 501 may also include onboard memory for caching purposes. The processor 501 may include a single processing unit or multiple processing units for performing different actions of the method flow according to an embodiment of this application.

[0119] RAM 503 stores various programs and data required for the operation of electronic device 500. Processor 501, ROM 502, and RAM 503 are interconnected via bus 504. Processor 501 executes various operations of the method flow according to embodiments of this application by executing programs in ROM 502 and / or RAM 503. It should be noted that the programs may also be stored in one or more memories other than ROM 502 and RAM 503. Processor 501 may also execute various operations of the method flow according to embodiments of this application by executing programs stored in said one or more memories.

[0120] According to embodiments of this application, the electronic device 500 may further include an input / output (I / O) interface 505, which is also connected to a bus 504. The electronic device 500 may also include one or more of the following components connected to the input / output (I / O) interface 505: an input section 506 including a keyboard, mouse, etc.; an output section 507 including a cathode ray tube (CRT), liquid crystal display (LCD), etc., and a speaker, etc.; a storage section 508 including a hard disk, etc.; and a communication section 509 including a network interface card such as a LAN card, modem, etc. The communication section 509 performs communication processing via a network such as the Internet. A drive 510 is also connected to the input / output (I / O) interface 505 as needed. A removable medium 511, such as a disk, optical disk, magneto-optical disk, semiconductor memory, etc., is installed on the drive 510 as needed so that computer programs read from it can be installed into the storage section 508 as needed.

[0121] This application also provides a computer-readable storage medium, which may be included in the device / apparatus / system described in the above embodiments; or it may exist independently and not assembled into the device / apparatus / system. The computer-readable storage medium carries one or more programs, which, when executed, implement the method according to the embodiments of this application.

[0122] According to embodiments of this application, the computer-readable storage medium can be a non-volatile computer-readable storage medium, such as including but not limited to: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof. In this application, the computer-readable storage medium can be any tangible medium containing or storing a program that can be used by or in conjunction with an instruction execution system, apparatus, or device. For example, according to embodiments of this application, the computer-readable storage medium may include ROM 502 and / or RAM 503 and / or one or more memories other than ROM 502 and RAM 503 described above.

[0123] Embodiments of this application also include a computer program product comprising a computer program containing program code for performing the methods shown in the flowchart. When the computer program product is run on a computer system, the program code enables the computer system to implement the material constitutive equation construction method based on a large language model provided in the embodiments of this application.

[0124] When the computer program is executed by the processor 501, it performs the functions defined in the system / apparatus of this application embodiment. According to the embodiments of this application, the systems, apparatuses, modules, units, etc., described above can be implemented by computer program modules.

[0125] In one embodiment, the computer program may rely on a tangible storage medium such as an optical storage device or a magnetic storage device. In another embodiment, the computer program may also be transmitted and distributed in the form of signals over a network medium, and may be downloaded and installed via the communication section 509, and / or installed from a removable medium 511. The program code contained in the computer program can be transmitted using any suitable network medium, including but not limited to: wireless, wired, etc., or any suitable combination thereof.

[0126] In such an embodiment, the computer program can be downloaded and installed from a network via communication section 509, and / or installed from removable medium 511. When the computer program is executed by processor 501, it performs the functions defined in the system of this application embodiment. According to embodiments of this application, the systems, devices, apparatuses, modules, units, etc., described above can be implemented by computer program modules.

[0127] According to embodiments of this application, program code for executing the computer programs provided in the embodiments of this application can be written in any combination of one or more programming languages. Specifically, these computational programs can be implemented using high-level procedural and / or object-oriented programming languages, and / or assembly / machine languages. Programming languages ​​include, but are not limited to, languages ​​such as Java, C++, Python, "C", or similar programming languages. The program code can be executed entirely on the user's computing device, partially on the user's device, partially on a remote computing device, or entirely on a remote computing device or server. In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0128] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of this application. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions indicated in the blocks may occur in a different order than those indicated in the drawings. For example, two consecutively indicated blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in a block diagram or flowchart, and combinations of blocks in a block diagram or flowchart, may be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.

[0129] Those skilled in the art will understand that the features described in the various embodiments of this application can be combined and / or combined in various ways, even if such combinations or combinations are not explicitly described in this application. In particular, the features described in the various embodiments of this application can be combined and / or combined in various ways without departing from the spirit and teachings of this application. All such combinations and / or combinations fall within the scope of this application.

[0130] The embodiments of this application have been described above. However, these embodiments are merely illustrative and not intended to limit the scope of this application. Although various embodiments have been described above, this does not mean that the measures in the various embodiments cannot be used advantageously in combination. Without departing from the scope of this application, those skilled in the art can make various substitutions and modifications, all of which should fall within the scope of this application.

Claims

1. A method for constructing material constitutive equations based on a large language model, characterized in that, The method includes: Obtain experimental datasets of stress and strain in materials; Based on the first major language model, data analysis is performed on the experimental dataset and the current set of equations to generate natural language strategies. The first major language model is used to extract variable coupling relationships that represent the dependencies between variables and structural pattern features that represent the regularity of the mathematical structure of the current set of equations. The natural language strategy is input into the second largest language model to output a set of candidate equations; The candidate equation set is evaluated by a multi-objective Pareto optimization of fitting error, generalization error and equation complexity, and the Pareto optimal equation set is updated by non-dominated sorting. The updated set of Pareto optimal equations is fed back to the first language model, and the policy generation and candidate equation generation operations are repeated until the preset convergence condition is met; and The Pareto optimal equation set is output to form the Pareto front of the material constitutive equation. The Pareto front is used to provide a set of candidate equation solutions that have a non-dominant relationship between fitting error, generalization ability and equation complexity. The multi-objective Pareto optimization evaluation of the candidate equation set, including fitting error, generalization error, and equation complexity, includes: For each equation in the candidate equation set, the fitting error index, generalization error index, and equation complexity index are calculated independently. The fitting error index, generalization error index, and equation complexity index are jointly constructed as a three-dimensional optimization objective. A non-dominated sorting strategy is used for solution set optimization. The sorting strategy determines whether any candidate equation is not greater than other equations in all three indices and whether the value of at least one index is less than the value of the corresponding index of other equations. The step of performing data analysis on the experimental dataset and the current set of equations based on the first large language model to generate a natural language strategy includes: Data analysis tools are used to extract variable coupling relationships from the experimental dataset to characterize the dependencies between variables. These data analysis tools include linear correlation analysis tools, nonlinear correlation analysis tools, time-frequency analysis tools, and causal inference tools. Specifically, the linear correlation analysis tool is used to extract the Pearson correlation coefficient between stress and strain; the nonlinear correlation analysis tool is used to extract the Spearman rank correlation coefficient between variables; the time-frequency analysis tool is used to identify the frequency components and periodic patterns in the stress-strain data; and the causal inference tool is used to determine the causal dependence of the strain history on the stress response. Abstract syntax tree parsing is performed on the current set of equations to obtain structural pattern features; and Natural language strategies are generated based on the variable coupling relationships and the structural pattern features.

2. The method according to claim 1, characterized in that, The process for determining a non-dominated sorting strategy includes: Two candidate equations are randomly selected and compared item by item using three indicators. If one equation is no greater than the other equation in terms of fitting error, generalization error and complexity, and at least one indicator is less than the corresponding indicator of the other equation, then the former is determined to dominate the latter. After traversing all candidate equations, equations that do not have any dominance relationship are retained in the Pareto optimal set.

3. The method according to claim 1, characterized in that, Updating the Pareto optimal equation set through non-dominated sorting includes: The candidate equation set and the historical Pareto optimal equations are merged into a temporary solution set; The temporary solution set is sorted non-dominated to remove dominated equations and update the current Pareto front.

4. A device for constructing material constitutive equations based on a large language model, characterized in that, The device includes: The acquisition module is used to acquire experimental datasets of stress and strain of materials; The natural language strategy generation module is used to perform data analysis on the experimental dataset and the current set of equations based on the first large language model to generate natural language strategies. The first large language model is used to extract variable coupling relationships that represent the dependencies between variables and structural pattern features that represent the regularity of the current set of equations in mathematical structure. The first output module is used to input the natural language strategy into the second large language model to output a set of candidate equations; The multi-objective evaluation module is used to perform multi-objective Pareto optimization evaluation of the candidate equation set, including fitting error, generalization error and equation complexity, and to update the Pareto optimal equation set through non-dominated sorting. The iterative update module is used to feed the updated Pareto optimal equation set back to the first language model, and repeatedly execute the policy generation and candidate equation generation operations until the preset convergence condition is reached. The second output module is used to output the Pareto optimal equation set as the Pareto front of the material constitutive equation. The Pareto front is used to provide a set of candidate equation solutions that have a non-dominant relationship between fitting error, generalization ability and equation complexity. The multi-objective evaluation module is also used to independently calculate the fitting error index, generalization error index, and equation complexity index for each equation in the candidate equation set; the fitting error index, generalization error index, and equation complexity index are jointly constructed into a three-dimensional optimization objective, and a non-dominated sorting strategy is used for solution set optimization. The sorting strategy determines whether any candidate equation is not greater than other equations in all three indices and whether the value of at least one index is less than the value of the corresponding index of other equations. The natural language strategy generation module is also used to invoke data analysis tools to extract variable coupling relationships from the experimental dataset to characterize the dependencies between variables. These data analysis tools include linear correlation analysis tools, nonlinear correlation analysis tools, time-frequency analysis tools, and causal inference tools. Specifically, the linear correlation analysis tool is used to extract the Pearson correlation coefficient between stress and strain; the nonlinear correlation analysis tool is used to extract the Spearman rank correlation coefficient between variables; the time-frequency analysis tool is used to identify frequency components and periodic patterns in the stress-strain data; and the causal inference tool is used to determine the causal dependence of the strain history on the stress response. The module performs abstract syntax tree parsing on the current equation set to obtain structural pattern features; and generates a natural language strategy based on the variable coupling relationships and the structural pattern features.

5. An electronic device, comprising: One or more processors; Memory, used to store one or more computer programs. The characteristic feature is that the one or more processors execute the one or more computer programs to implement the steps of the method according to any one of claims 1 to 3.

6. A computer-readable storage medium having a computer program or instructions stored thereon, characterized in that, When the computer program or instructions are executed by a processor, they implement the steps of the method according to any one of claims 1 to 3.

7. A computer program product, comprising a computer program or instructions, characterized in that, When the computer program or instructions are executed by a processor, they implement the steps of the method according to any one of claims 1 to 3.

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