A method for constructing a proxy model of structural strength analysis

Abstract

The application discloses a method for constructing a surrogate model for structural strength analysis, and relates to the technical field of computers, which comprises the following steps: obtaining an initial training set containing finite element simulation results of a plurality of design samples, wherein the simulation results comprise full-field stress-strain distribution data and scalar strength response data; constructing a parallel physical information neural network module and a Kriging module for separate training, wherein the loss function of the physical information neural network module comprises a data fitting loss term and a physical constraint loss term, and the Kriging module outputs scalar strength response and prediction variance; performing an iterative operation until convergence: selecting a plurality of candidate design points from unsampled design points, calculating a comprehensive utility function value by fusing prediction variance and physical equation residual, selecting a candidate design point corresponding to a maximum value as a supplementary sample point, and adding supplementary data to the training set after simulation and retraining the two modules. The technical scheme improves the evaluation capability of the surrogate model.

Description

Technical Field

[0001] This application relates to the field of computer technology, specifically to a method for constructing a surrogate model for structural strength analysis. Background Technology

[0002] In structural optimization design, high-precision finite element simulation is commonly used for static strength analysis. However, for complex structures, single simulation calculations are costly and time-consuming, making it difficult to meet the needs of large-scale optimization or real-time evaluation. Surrogate model technology, by constructing low-computational-cost approximate models to replace time-consuming simulations, has become an effective means of solving these problems.

[0003] Existing surrogate model methods, such as single Kriging models or traditional neural networks, suffer from the following problems: First, with a small initial sample size, the model's prediction accuracy is insufficient and its generalization ability is poor. Second, purely data-driven models lack physical constraints, and the prediction results may violate fundamental physical laws such as mechanical equilibrium, lacking interpretability. Furthermore, existing surrogate models struggle to balance full-field response prediction with uncertainty quantification; sample selection relies on manual or random strategies, resulting in low sample utilization efficiency and failing to meet the requirements for efficient and high-precision structural strength assessment. Therefore, how to construct a structural static strength surrogate model with both high accuracy and physical consistency under limited sample conditions is a pressing technical problem to be solved in this field. Summary of the Invention

[0004] To address the aforementioned technical problems, this application provides a method for constructing a surrogate model for structural strength analysis.

[0005] In a first aspect, this application provides a method for constructing a surrogate model for structural strength analysis, comprising: obtaining an initial training set, which includes finite element simulation results corresponding to multiple design samples, wherein the finite element simulation results include full-field stress-strain distribution data and scalar strength response data; constructing a parallel physical information neural network module and a Kriging module, and training the physical information neural network module and the Kriging module respectively using the initial training set, wherein the physical information neural network module is used to output the predicted full-field stress-strain distribution based on the input design variables and spatial coordinates, and the loss function of the physical information neural network module includes a data fitting loss term and a physical constraint loss term, and the Kriging module is used to output the predicted scalar strength response based on the input design variables and provide the prediction variance; performing the following iterative operations until the preset convergence condition is met to obtain the target hybrid surrogate model: Multiple candidate design points are selected from the unsampled design points within the design space. The comprehensive utility function value of the multiple candidate design points is calculated. The candidate design points are different from the design points sampled in the initial training. The comprehensive utility function is obtained by fusing the first index and the second index. The first index is the prediction variance of the Kriging module for the multiple candidate design points. The second index is the physical equation residual obtained by calculating the full-field stress-strain distribution of the multiple candidate design points based on the prediction of the physical information neural network module. The candidate design point with the largest comprehensive utility function value is selected as a supplementary sample point. Finite element simulation is performed on the supplementary sample point to obtain supplementary data. The supplementary data is added to the initial training set to update the training set. The physical information neural network module and the Kriging module are retrained based on the updated training set. It is determined whether the convergence condition is met. If it is met, the iteration operation is stopped. If it is not met, the next round of iteration operation is continued.

[0006] By adopting the above technical solution, an initial training set containing full-field stress-strain distribution data and scalar strength response data is obtained, providing a data foundation for model training. A parallel physical information neural network module and a Kriging module are constructed, which can predict the full-field stress-strain distribution and scalar strength response respectively. Furthermore, the loss function of the physical information neural network module includes data fitting loss terms and physical constraint loss terms, ensuring that the prediction results conform to physical laws. Through iterative operations, supplementary sample points are selected based on the comprehensive utility function value to update the training set and retrain the module. This solves the technical problems of low prediction accuracy, poor physical consistency, and low sample utilization efficiency of surrogate models in related technologies with a small number of initial samples. It improves the model's prediction accuracy and generalization ability, ultimately obtaining a target hybrid surrogate model that meets the requirements for efficient and high-precision structural strength assessment, thus enhancing the evaluation capability of the surrogate model.

[0007] Optionally, the loss function for the physical information neural network module can take the following form: ;in, Let θ be the loss function of the physical information neural network module, and let θ be the set of network parameters of the physical information neural network module. For data fitting loss term, For the physical constraint loss term, λ D λ represents the weighting coefficient of the data fitting loss term. P This represents the weighting coefficient of the physical constraint loss term.

[0008] By adopting the above technical solution, the loss function of the physical information neural network module is set as a weighted sum of the data fitting loss term and the physical constraint loss term. This balances data fitting and physical constraints, ensuring that the prediction results closely resemble finite element simulation results while also satisfying the elasticity control equations. This improves the model's prediction accuracy and physical consistency, addressing the problem of existing purely data-driven models lacking physical constraints and potentially violating fundamental physical laws in their predictions. Furthermore, the model maintains mechanical consistency even with small sample sizes, enhancing prediction reliability and engineering usability.

[0009] Optional, data fitting loss term Used to measure the deviation between the predicted value and the corresponding point in the finite element simulation result; data fitting loss term. Adopt the following form: , where N d x is the number of sample points used for supervised training. i Let v be the spatial coordinates of the i-th sampling point. i Let i be the design variable vector corresponding to the i-th sampling point. For the displacement field predicted by the physical information neural network module, u i The displacement reference value corresponding to the i-th sampling point is obtained from the finite element simulation. The stress field predicted by the physical information neural network module, σ i The stress reference value corresponding to the i-th sampling point is derived from the finite element simulation; physical constraint loss term. Used to force prediction results to satisfy the elasticity control equations, physical constraint loss term. Adopt the following form: , where N c x represents the number of randomly sampled physical constraint placement points within the structural geometry domain. j Let v be the spatial coordinates of the j-th sampling point in the randomly sampled physical constraint configuration points. j Let be the design variable vector corresponding to the j-th sampling point in the randomly sampled physical constraint configuration points, and let b be the body force term. Let ε be the fourth-order elastic stiffness tensor of the material, and ε be the displacement based on the predicted displacement. The strain tensor is obtained by calculating the geometric equation.

[0010] By adopting the above technical solutions, using a specific form of data fitting loss term to measure the deviation between the predicted value and the corresponding point of the finite element simulation result, the predicted displacement field and stress field of the physical information neural network module can be closer to the reference value of the finite element simulation, thus improving the prediction accuracy. By using a specific form of physical constraint loss term to force the prediction result to satisfy the elasticity control equation, the prediction result can conform to physical laws, enhance the physical consistency of the model, and overcome the problem of the lack of physical constraints in the pure data-driven model.

[0011] Optionally, the loss function of the physical information neural network module adopts an adaptive weighting strategy, dynamically adjusting the weight coefficients of the data fitting loss term and the physical constraint loss term during training.

[0012] By adopting the above technical solution, an adaptive weighting strategy is used for the loss function of the physical information neural network module. During the training process, the weight coefficients of the data fitting loss term and the physical constraint loss term are dynamically adjusted, which enables the physical information neural network module to better balance data fitting and physical constraints during training, thereby further improving the prediction accuracy and physical consistency of the model and meeting the requirements of efficient and high-precision structural strength assessment.

[0013] Optionally, the adaptive weighting strategy includes: during the training of the physical information neural network module, after every preset number of training epochs, calculating the ratio of the gradient norm of the data fitting loss to the gradient norm of the physical constraint loss. and according to λ P =γ·α1 Dynamically adjust the weight coefficient λ of the physical constraint loss term P , and λ D =1, where α1 is the preset initial physical weight coefficient.

[0014] By adopting the above technical solution, during the training process of the physical information neural network module, after a preset number of training cycles, the weight coefficient of the physical constraint loss term is dynamically adjusted according to the ratio of the gradient norm of the data fitting loss to the gradient norm of the physical constraint loss. This makes the weights of the data fitting loss term and the physical constraint loss term in the loss function more reasonable, thereby improving the training effect of the physical information neural network module. This allows the model to better meet physical constraints while taking into account data fitting, improving the model's accuracy and physical consistency, and ultimately improving the surrogate model's accuracy in evaluating the static strength of the structure.

[0015] Optionally, the comprehensive utility function may combine the first and second indicators in the following form: ,in, For design points that were not sampled, The first indicator after normalization. The second index is the normalized index, and α2 is the balancing weight coefficient, α2∈[0,1]; the first index Adopt the following form: , where σ Z ² represents the process variance, R is the correlation matrix of the Kriging model, r is the correlation vector between the design point to be tested and the existing sample points, and 1 represents the unit column vector; the second index Adopt the following form: , where N c x represents the number of physical constraint placement points randomly sampled within the structural geometry domain. k Let θ be the spatial coordinates of the k-th configuration point, and θ be the set of network parameters for the physical information neural network module. This represents the mean of the equilibrium equation residuals of the physical information neural network modules on the training set after initial training. This represents the mean constitutive residual of the physical information neural network module on the training set after initial training.

[0016] By adopting the above technical solution and integrating the first and second indicators using a comprehensive utility function, supplementary sample points can be determined based on the prediction variance of the Kriging module and the physical equation residuals of the physical information neural network module. This effectively improves sample utilization efficiency and constructs a structural static strength surrogate model with both high accuracy and physical consistency under limited sample conditions, meeting the requirements for efficient and high-precision structural strength assessment. Furthermore, the specific forms of the first and second indicators are provided, making the calculation of the comprehensive utility function more accurate and reasonable, which helps to select supplementary sample points more precisely and improve the performance of the surrogate model.

[0017] Optional, the first indicator after normalization and the normalized second indicator The results are obtained using the following formulas: , ,in, and These represent the maximum and minimum values ​​of the first index among multiple candidate design point sets. and These are the maximum and minimum values ​​of the second index in a set of multiple candidate design points, respectively, where δ is a preset positive constant to prevent the denominator from being zero.

[0018] By adopting the above technical solution and normalizing the first and second indicators, the calculation of the comprehensive utility function is avoided due to differences in data range, the accuracy of selecting supplementary sample points is improved, and thus it is conducive to building a more efficient and accurate target hybrid proxy model.

[0019] Optionally, the balancing weights are dynamically adjusted in each iteration as follows: ,in, Let Variance be the variance of the first index across multiple candidate design point sets. The variance of the second index over a set of multiple candidate design points.

[0020] By adopting the above technical solution, and by dynamically adjusting the balancing weight coefficients according to a specific formula in each iteration, the weights of the first indicator (the variance predicted by the Kriging module) and the second indicator (the residual of the physical equation predicted by the physical information neural network module) in the comprehensive utility function can be adaptively balanced based on the variances of the first and second indicators on multiple candidate design point sets. This allows the comprehensive utility function to more reasonably reflect the value of the candidate design points, thereby more accurately selecting supplementary sample points, improving the efficiency of sample utilization during model construction, and helping to construct a structural static strength surrogate model that combines high accuracy and physical consistency.

[0021] Optionally, the convergence condition includes at least one of the following: the number of iterations reaches a preset upper limit; the maximum value of the comprehensive utility function is less than a first preset threshold; the change in the data fitting loss of the physical information neural network module within a preset number of consecutive iterations is less than a preset loss threshold, and the residual of the physical equation corresponding to the physical information neural network module is less than a preset residual threshold.

[0022] By adopting the above technical solutions, multiple selectable convergence conditions are provided to terminate the iteration operation. When the number of iterations reaches the preset upper limit, infinite iteration can be avoided, saving computational resources. When the maximum value of the comprehensive utility function is less than the first preset threshold, it indicates that the benefits brought by supplementing samples are limited, and stopping the iteration can improve efficiency. When the data fitting loss of the physical information neural network module changes little and the physical equation residual is also small, it indicates that the model has reached a certain accuracy. At this time, stopping the iteration can ensure that the obtained target hybrid proxy model has good performance.

[0023] Optionally, the Kriging module employs a general Kriging model, whose regression basis function includes the scalar intensity response predicted by the physical information neural network module as an auxiliary variable; the general Kriging model takes the following form: Where v is the current design variable vector, y(v) is the scalar intensity response prediction value of the scheme corresponding to the current design variable v by the Kriging module, β0 is the constant term, and β1 is the regression coefficient. Z(v) represents the maximum value of the full-field stress-strain distribution predicted by the physical information neural network module for the current design variable v, and Z(v) represents the Gaussian process component.

[0024] By adopting the above technical solution, the Kriging module uses a general Kriging model, and the regression basis function includes the scalar intensity response predicted by the physical information neural network module as an auxiliary variable, which can improve the prediction accuracy of the Kriging module for the scalar intensity response. Combined with the prediction results of the physical information neural network module, the prediction of the Kriging module is more accurate and reliable, and better adapts to the needs of actual engineering applications.

[0025] Optionally, the above method also includes the following application steps: using the target hybrid surrogate model for static strength assessment at the new design point, outputting the predicted full-field stress-strain distribution at the new design point through the physical information neural network module, and outputting the predicted scalar strength response and the confidence interval of the predicted scalar strength response at the new design point through the Kriging module.

[0026] By adopting the above technical solution, the target hybrid surrogate model can be applied to the static strength assessment of the new design point. The physical information neural network module outputs the predicted full-field stress and strain distribution corresponding to the new design point, and the Kriging module outputs the predicted scalar strength response and the confidence interval of the predicted scalar strength response corresponding to the new design point, thereby achieving efficient assessment of the static strength of the structure.

[0027] In a second aspect of this application, a surrogate model construction apparatus for structural strength analysis is also provided, used to execute the surrogate model construction method for structural strength analysis according to any of the preceding claims, comprising: an acquisition module for acquiring an initial training set, the initial training set containing finite element simulation results corresponding to multiple design samples, wherein the finite element simulation results include full-field stress-strain distribution data and scalar strength response data; a construction module for constructing a parallel physical information neural network module and a Kriging module, and training the physical information neural network module and the Kriging module respectively using the initial training set, wherein the physical information neural network module is used to output the predicted full-field stress-strain distribution based on input design variables and spatial coordinates, the loss function of the physical information neural network module includes a data fitting loss term and a physical constraint loss term, the Kriging module is used to output the predicted scalar strength response based on input design variables and provide the prediction variance; and an iteration module for performing the following iterative operations. The process continues until a preset convergence condition is met to obtain the target hybrid surrogate model: Multiple candidate design points are selected from the unsampled design points in the design space, and the comprehensive utility function value of these candidate design points is calculated. The candidate design points differ from the sampled design points in the initial training. The comprehensive utility function is obtained by fusing a first index and a second index. The first index is the prediction variance of the Kriging module for the multiple candidate design points; the second index is the physical equation residual obtained from the full-field stress-strain distribution calculation of the multiple candidate design points based on the prediction of the physical information neural network module. The candidate design point with the largest comprehensive utility function value is selected as a supplementary sample point. Finite element simulation is performed on the supplementary sample point to obtain supplementary data. This supplementary data is added to the initial training set to update the training set, and the physical information neural network module and the Kriging module are retrained based on the updated training set. The convergence condition is then determined. If it is met, the iteration operation stops; otherwise, the next iteration operation continues.

[0028] In a third aspect of this application, an electronic device is also provided, including a memory and a processor, wherein a computer program is stored in the memory, and the processor executes the program to implement the method steps of any of the above claims.

[0029] In a fourth aspect of this application, a computer-readable storage medium is also provided, which stores instructions that, when executed, perform the method steps of any of the above claims.

[0030] In summary, one or more technical solutions provided in this application have at least the following technical effects or advantages: 1. A parallel physical information neural network module and a Kriging module are constructed to predict the full-field stress-strain distribution and scalar strength response, respectively. The loss function of the physical information neural network module includes data fitting loss terms and physical constraint loss terms, which can ensure that the prediction results conform to physical laws. Through iterative operations, supplementary sample points are selected to update the training set and retrain the module based on the comprehensive utility function value. This solves the technical problems of low prediction accuracy, poor physical consistency and low sample utilization efficiency of surrogate models in related technologies with a small number of initial samples. It can improve the prediction accuracy and generalization ability of the model, and finally obtain the target hybrid surrogate model, which meets the requirements of efficient and high-precision structural strength assessment and improves the assessment capability of the surrogate model. 2. Dynamically adjusting the weight coefficients of the data fitting loss term and the physical constraint loss term during training enables the physical information neural network module to better balance data fitting and physical constraints during training, further improving the model's prediction accuracy and physical consistency, so as to meet the needs of efficient and high-precision structural strength assessment. Attached Figure Description

[0031] Figure 1 This is a flowchart of a surrogate model construction method for structural strength analysis provided in an embodiment of this application; Figure 2 This is a structural block diagram of a proxy model construction device for structural strength analysis provided in an embodiment of this application; Figure 3 This is a schematic diagram of the structure of an electronic device disclosed in an embodiment of this application.

[0032] Explanation of reference numerals in the attached figures: 300 - Electronic device; 301 - Processor; 302 - Communication bus; 303 - User interface; 304 - Network interface; 305 - Memory. Detailed Implementation

[0033] To enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this specification will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments.

[0034] In the description of the embodiments of this application, the words "for example" or "for instance" are used to indicate examples, illustrations, or explanations. Any embodiment or design that is described as "for example" or "for instance" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or design options. Rather, the use of the words "for example" or "for instance" is intended to present the relevant concepts in a specific manner.

[0035] In the description of the embodiments of this application, the term "multiple" means two or more. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. The terms "comprising," "including," "having," and variations thereof all mean "including but not limited to," unless otherwise specifically emphasized.

[0036] This application provides a method for constructing a surrogate model for structural strength analysis, such as... Figure 1 As shown, Figure 1 This is a flowchart of a surrogate model construction method for structural strength analysis provided in an embodiment of this application, including the following steps: Step S101: Obtain the initial training set, which contains finite element simulation results corresponding to multiple design samples. The finite element simulation results include full-field stress-strain distribution data and scalar strength response data. Step S102: Construct a parallel physical information neural network module and a Kriging module, and train the physical information neural network module and the Kriging module respectively using the initial training set. The physical information neural network module is used to output the predicted full-field stress and strain distribution based on the input design variables and spatial coordinates. The loss function of the physical information neural network module includes a data fitting loss term and a physical constraint loss term. The Kriging module is used to output the predicted scalar strength response based on the input design variables and provide the prediction variance. Step S103: Perform the following iterative operations until the preset convergence condition is met to obtain the target hybrid surrogate model: Select multiple candidate design points from the unsampled design points in the design space, calculate the comprehensive utility function value of the multiple candidate design points, wherein the candidate design points are different from the design points sampled in the initial training, and the comprehensive utility function is obtained by fusing the first index and the second index; the first index is the prediction variance of the Kriging module for the multiple candidate design points; the second index is the physical equation residual obtained by calculating the full-field stress-strain distribution of the multiple candidate design points based on the prediction of the physical information neural network module; select the candidate design point with the largest comprehensive utility function value as the supplementary sample point; perform finite element simulation on the supplementary sample point to obtain supplementary data, add the supplementary data to the initial training set to update the training set, and retrain the physical information neural network module and the Kriging module based on the updated training set; determine whether the convergence condition is met, if it is met, stop the iterative operation, if it is not met, continue to perform the next round of iterative operation.

[0037] Through the above steps, an initial training set containing full-field stress-strain distribution data and scalar strength response data is obtained, providing a data foundation for model training. A parallel physical information neural network module and a Kriging module are constructed, which can predict the full-field stress-strain distribution and scalar strength response respectively. The loss function of the physical information neural network module includes data fitting loss terms and physical constraint loss terms, ensuring that the prediction results conform to physical laws. Through iterative operations, supplementary sample points are selected based on the comprehensive utility function value to update the training set and retrain the module. This solves the technical problems of low prediction accuracy, poor physical consistency, and low sample utilization efficiency of surrogate models in related technologies with a small number of initial samples. It improves the model's prediction accuracy and generalization ability, ultimately resulting in a target hybrid surrogate model that meets the requirements for efficient and high-precision structural strength assessment, thus enhancing the evaluation capability of the surrogate model.

[0038] This embodiment constructs a parallel heterogeneous hybrid proxy model architecture based on "physical mechanism + data-driven" principles. Its core principle lies in breaking the limitations of traditional single proxy models by deploying a Physical Information Neural Network (PINN) and a Kriging model in parallel to handle different types of data modes: the PINN module is responsible for learning the continuous full-field stress-strain distribution (tensor field data), while the Kriging module is responsible for capturing the statistical characteristics and uncertainties of discrete scalar strength responses (such as maximum stress). The two are coupled through a dual-driven active learning loop, which utilizes a comprehensive utility function that integrates "statistical uncertainty (Kriging variance)" and "physical inconsistency (PINN physical residuals)" to intelligently guide the adaptive expansion of high-cost finite element simulation samples. This method solves the problems in related technologies where a single model cannot simultaneously address "physical constraints" and "quantification of prediction uncertainty," as well as the problems of poor model accuracy and slow convergence under small sample conditions. It significantly reduces the number of calls to high-fidelity simulations and achieves rapid and high-precision evaluation of structural static strength while ensuring that the prediction results conform to the fundamental laws of mechanics. This embodiment achieves functional complementarity by paralleling a neural network embedded with physical laws (PINN) and a Kriging model that provides statistical predictions. Furthermore, a dual-driven adaptive sampling mechanism integrating "statistical uncertainty" and "physical violation degree" actively selects the most valuable sample points for supplementary simulations, improving model accuracy with minimal simulation iterations. Specifically, an initial training set is first obtained, containing full-field stress cloud map data (stress values ​​at each node) and scalar strength data (such as maximum stress values) obtained from high-precision finite element simulations of a few design samples. For example, the full-field stress cloud map data includes the coordinates of 5000 (or 10,000) nodes on a structural component and their corresponding stress values, while the scalar strength response data is the maximum equivalent stress value. Each design sample corresponds to a working condition of the structural component (with fixed geometry and thickness), such as being subjected to different external forces at a certain location. Each design sample contains full-field stress-strain data and scalar strength response data obtained from finite element simulations. Then, two parallel modules are constructed: the PINN module takes design variables and spatial coordinates as input and outputs the predicted stress value at that point. Its loss function penalizes both the deviation of the predicted value from the actual simulation value (data fitting) and the behavior of the predicted value violating the mechanical equilibrium equations (physical constraints), thus ensuring that the prediction results conform to physical laws even in places where there is no data; the Kriging module takes design variables as input and directly outputs the predicted scalar strength (such as maximum stress) and its prediction variance (indicating the confidence level in the prediction). Initial training can stop when the loss value of PINN is less than a preset threshold.In the iterative update step, the system randomly or sampled multiple candidate design points from unsampled points in the design space. For each candidate point, a comprehensive utility function is calculated, which integrates two indicators: first, the Kriging prediction variance (representing statistical uncertainty; a large variance indicates insufficient data and unreliable predictions in that area); and second, the physical equation residual calculated based on the full-field stress distribution predicted by PINN for that candidate design point (representing the degree of physical violation; a large residual indicates that PINN severely violates the laws of mechanics at that design point). The candidate design point with the largest comprehensive utility function value is selected as a supplementary sample point, and a high-precision finite element simulation is performed on it. The new data is added to the training set, and both modules are retrained. This process is repeated until the model accuracy meets the requirements. Optionally, within the current design variable space, a sufficient set of candidate points is generated using the Latin hypercube sampling method. These candidate points are distributed throughout the entire design domain but do not include existing training sample points to ensure global exploratory nature. This method, by simultaneously focusing on "statistical unknowns" and "physical contradictions," ensures that each supplementary simulation brings maximum information gain, thereby constructing a high-precision, physically reliable surrogate model with minimal samples. This embodiment addresses the problems of overfitting and poor generalization ability of existing surrogate models with small sample sizes by actively supplementing the most valuable samples through adaptive iteration. It also solves the problem of pure data-driven models violating physical laws through the physical constraint loss term of PINN. Furthermore, it addresses the difficulty of existing surrogate models in simultaneously considering full-field response prediction and uncertainty quantification, as well as low sample utilization efficiency, through the PINN+Kriging parallel architecture and dual-driven sampling. This embodiment allows for the construction of a high-precision surrogate model (target hybrid surrogate model) using 20-50 initial samples and 5-10 iterations. The model can output full-field stress contour maps and key scalar strengths and their confidence intervals in milliseconds. The prediction results automatically satisfy the mechanical equilibrium equations and have physical interpretability.

[0039] The scalar strength response in the above embodiments includes the maximum equivalent stress of the structure or the safety factor, and the stress-strain distribution across the entire field includes stress cloud map data or displacement field data.

[0040] In an optional embodiment, the loss function of the physical information neural network module takes the following form: ;in, Let θ be the loss function of the physical information neural network module, and let θ be the set of network parameters of the physical information neural network module. For data fitting loss term, For the physical constraint loss term, λ D λ represents the weighting coefficient of the data fitting loss term. P This represents the weighting coefficient of the physical constraint loss term.

[0041] In this embodiment, the loss function of the physical information neural network module is set as a weighted sum of data fitting loss and physical constraint loss. This balances data fitting and physical constraints, ensuring that the prediction results closely resemble finite element simulation results while also satisfying the elasticity control equations. This improves the model's prediction accuracy and physical consistency, addressing the problem of existing purely data-driven models lacking physical constraints and potentially violating fundamental physical laws in their predictions. Furthermore, it allows the model to maintain mechanical consistency even with small sample sizes, enhancing prediction reliability and engineering usability.

[0042] In traditional neural network training, the loss function typically only includes a data fitting term (such as mean squared error), meaning it only requires the network output to be as close as possible to the true values ​​of the training samples. This approach works reasonably well when there are sufficient samples, but it is prone to overfitting when the samples are sparse. Furthermore, the network is completely unconstrained by physical laws and may output any result that satisfies the data distribution but is physically absurd. For example, a PINN trained only on data might predict non-zero stress divergence in regions without external forces, violating the mechanical equilibrium equations. This embodiment introduces a physical constraint loss term. and compare it with the data fitting loss. Weighted summation ensures that the network is simultaneously constrained by two objectives during training: "fitting the data" and "obeying the laws of physics." The data fitting loss guarantees that the predicted values ​​closely approximate the true values ​​of the finite element method, while the physical constraint loss forces the output to satisfy the equations of elasticity. D and λ P These are two weighting coefficients used to balance their impact on network parameter updates. When the initial sample size is very small, λ can be appropriately increased. P (Physical weights) allow the network to rely more on physical laws for reasoning, rather than forcibly fitting sparse and potentially noisy data. This design enables PINN to output physically reasonable predictions even in sparse data regions, thus solving the problem of poor generalization ability and physically unreliable predictions in pure data-driven models with small sample sizes. The trained PINN model not only performs well on known data points, but also makes more reliable predictions that conform to physical logic in unseen regions, significantly enhancing the model's generalization ability and physical interpretability.

[0043] In an optional embodiment, the data fitting loss term Used to measure the deviation between the predicted value and the corresponding point in the finite element simulation result; data fitting loss term. Adopt the following form: , where N d x is the number of sample points used for supervised training. i Let v be the spatial coordinates of the i-th sampling point. i Let i be the design variable vector corresponding to the i-th sampling point. For the displacement field predicted by the physical information neural network module, u i The displacement reference value corresponding to the i-th sampling point is obtained from the finite element simulation. The stress field predicted by the physical information neural network module, σ i The stress reference value corresponding to the i-th sampling point is derived from the finite element simulation; physical constraint loss term. Used to force prediction results to satisfy the elasticity control equations, physical constraint loss term. Adopt the following form: , where N c x represents the number of randomly sampled physical constraint placement points within the structural geometry domain. j Let v be the spatial coordinates of the j-th sampling point in the randomly sampled physical constraint configuration points. j Let be the design variable vector corresponding to the j-th sampling point in the randomly sampled physical constraint configuration points, and let b be the body force term. Let ε be the fourth-order elastic stiffness tensor of the material, and ε be the displacement based on the predicted displacement. The strain tensor is obtained by calculating the geometric equation.

[0044] In this embodiment, using a specific form of data fitting loss term to measure the deviation between the predicted value and the corresponding point of the finite element simulation result can make the predicted displacement field and stress field of the physical information neural network module closer to the reference value of the finite element simulation, thereby improving the prediction accuracy. Using a specific form of physical constraint loss term to force the prediction result to satisfy the elasticity control equation can make the prediction result conform to physical laws, enhance the physical consistency of the model, and overcome the problem of the lack of physical constraints in the pure data-driven model.

[0045] Data fitting loss term The joint mean square error of the displacement and stress fields is adopted, which includes two summation terms: the first term measures the deviation between the predicted displacement field and the finite element simulation displacement reference value, and the second term measures the deviation between the predicted stress field and the finite element simulation stress reference value; optionally, in practical applications, only the second term, i.e., the deviation value of the predicted stress field, can be used. Physical constraint loss term. It includes two key physical penalty terms: the first term forces the sum of the divergence of the predicted stress field and the body forces to be zero, which is precisely the equilibrium equation in elasticity; the second term forces the predicted stress to be consistent with the constitutive stress calculated through predicted displacement, i.e., satisfying the generalized Hooke's law. The physical constraint loss calculates the equilibrium equation residuals and constitutive equation residuals at the placement points, forcing the model output to satisfy the fundamental equations of physical laws. For example, the physical constraint placement points can be boundary sampling points. These constraints are randomly sampled N times throughout the entire structural geometry. cForced at a configuration point, where the configuration point does not necessarily have a finite element simulation value, and the number can be greater than the number of sample points (e.g., 5000, 3000, or others), for example, N. c =10000, and N d =5000×20=100000, thus enabling global physical constraints on model behavior at extremely low cost. This design cleverly integrates "supervised learning" (fitting data) with "unsupervised physical constraints" (enforcing adherence to equations) into a unified loss function. During backpropagation, the gradient originates from both data errors and physical residuals, driving its parameters to update in a direction that simultaneously minimizes both. This allows PINN to make reasonable predictions guided by physical laws even in sparse data regions, fundamentally solving the problems of physical inconsistency and poor generalization ability in purely data-driven models. This embodiment provides global supervision through physical constraints at placement points, solving the problem of uncontrollable predictions in data-free regions by purely data-driven models; it optimizes both simultaneously through a joint loss function, solving the problem of loose coupling between displacement and stress fields in existing PINN implementations; and it addresses the difficulty in ensuring that prediction results simultaneously satisfy equilibrium equations and constitutive relations in related technologies through constraints from two independent physical residual terms.

[0046] In an optional embodiment, the loss function of the physical information neural network module adopts an adaptive weighting strategy, dynamically adjusting the weight coefficients of the data fitting loss term and the physical constraint loss term during training.

[0047] In this embodiment, an adaptive weighting strategy is adopted for the loss function of the physical information neural network module. During the training process, the weight coefficients of the data fitting loss term and the physical constraint loss term are dynamically adjusted. This enables the physical information neural network module to better balance data fitting and physical constraints during training, further improving the prediction accuracy and physical consistency of the model, so as to meet the requirements of efficient and high-precision structural strength assessment.

[0048] During the training of PINN, fixed weight coefficients (λ) are no longer used. D and λ P Instead of using fixed weights, these weights are allowed to dynamically adjust themselves. This strategy aims to address a challenging technical problem: the convergence speed and magnitude of the data fitting loss and physical constraint loss can vary significantly at different stages of training. Using fixed weights can easily lead to one loss function becoming dominant, causing model training instability or getting stuck in local optima, ultimately affecting model accuracy. This problem is particularly pronounced in scenarios with few samples. Initially, the training sample size is small. It may be very unstable if λ P If λ is too large, the network will ignore this small amount of valuable data and only focus on satisfying the physical equations, resulting in predictions that are physically consistent but deviate significantly from the true values; if λ is too large...D If the sample size is too large, the network will memorize a small number of samples, completely disregarding physical laws, leading to overfitting and physical absurdity. This application addresses this by dynamically adjusting the weight coefficients during training to maintain a balance in the gradient contributions of the two losses. Through adaptive weighting, the system can automatically balance the training progress of the two tasks; for example, when the physical loss decreases slowly, its weight can be appropriately increased, and vice versa. This makes the training process more robust and efficient, better coordinating the two learning objectives of data-driven and physics-driven approaches. This strategy significantly improves the effectiveness of few-sample learning, ultimately accelerating model convergence, improving the prediction accuracy of the final model, and enhancing the stability of the training process.

[0049] In an optional embodiment, the adaptive weighting strategy includes: during the training process of the physical information neural network module, after every preset number of training cycles, calculating the ratio of the data fitting loss gradient norm to the physical constraint loss gradient norm. and according to λ P =γ·α1 Dynamically adjust the weight coefficient λ of the physical constraint loss term P , and λ D =1, where α1 is the preset initial physical weight coefficient.

[0050] In this embodiment, during the training process of the physical information neural network module, after a preset number of training cycles, the weight coefficient of the physical constraint loss term is dynamically adjusted according to the ratio of the gradient norm of the data fitting loss to the gradient norm of the physical constraint loss. This makes the weights of the data fitting loss term and the physical constraint loss term in the loss function more reasonable, thereby improving the training effect of the physical information neural network module. This allows the model to better meet physical constraints while taking into account data fitting, improving the model's accuracy and physical consistency, and thus improving the surrogate model's accuracy in evaluating the static strength of the structure.

[0051] This embodiment provides a specific implementation of an adaptive weighting strategy based on dynamic adjustment of the gradient norm. This is an efficient and easy-to-implement technical solution to ensure a balance between data fitting loss and physical constraint loss during PINN training. Its core idea is that during neural network training, the influence of a loss term on network parameters can be measured by the gradient norm of that loss term relative to the parameters. The larger the gradient norm, the more "powerful" the loss term is in the current parameter update. Ideally, we want both data fitting and physical constraints to have sufficient influence on parameter updates, rather than one dominating. Specifically, during training, the norm of the data loss gradient is calculated every preset number of training cycles (e.g., every 100 iterations). and the norm of the physical loss gradient Then calculate the ratio γ of the two. If γ is very small (e.g., 0.0001), it means that the gradient of the data loss is much smaller than the physical loss, implying that the physical loss dominates the parameter update and the data information is suppressed. In this case, according to λ... P =γ×α1 adjusts the weight of the physical loss. Since γ is very small, λ P It will also become smaller, thus reducing the impact of physical loss and giving data loss a greater voice. Conversely, if γ is large, then λ P Increase its size to strengthen physical constraints. The ultimate goal is to make λ larger. P × and They are on the same order of magnitude. λ D The value is fixed at 1 as a reference baseline. This method is simple and effective, requiring no additional training or complex optimizers, and is easily embedded into existing deep learning frameworks. This embodiment automatically maintains the gradients of data loss and physical loss on the same order of magnitude, resulting in a smoother training process; it reduces the workload of hyperparameter tuning, improving the ease of use and robustness of the method; it is particularly suitable for small sample scenarios, ensuring that physical knowledge appropriately guides training without suppressing the data.

[0052] In an optional embodiment, the combined utility function integrates the first and second metrics in the following form: ,in, Design points that were not sampled The first indicator after normalization. The second index is the normalized index, and α2 is the balancing weight coefficient, α2∈[0,1]; the first index Adopt the following form: , where σ Z ² represents the process variance, R is the correlation matrix of the Kriging model, r is the correlation vector between the design point to be tested and the existing sample points, and 1 represents the unit column vector; the second index Adopt the following form: , where N c x represents the number of physical constraint placement points randomly sampled within the structural geometry domain. k Let θ be the spatial coordinates of the k-th configuration point, and θ be the set of network parameters for the physical information neural network module. This represents the mean of the equilibrium equation residuals of the physical information neural network modules on the training set after initial training. This represents the mean constitutive residual of the physical information neural network module on the training set after initial training.

[0053] In this embodiment, by fusing the first and second indices using a comprehensive utility function, supplementary sample points can be determined based on the prediction variance of the Kriging module and the physical equation residuals of the physical information neural network module. This effectively improves sample utilization efficiency and constructs a structural static strength surrogate model with both high accuracy and physical consistency under limited sample conditions, meeting the requirements for efficient and high-precision structural strength assessment. Furthermore, the specific forms of the first and second indices are provided, making the calculation of the comprehensive utility function more accurate and reasonable, which helps to select supplementary sample points more precisely and improve the performance of the surrogate model.

[0054] This embodiment provides the specific mathematical form of the comprehensive utility function in adaptive sampling, which is the core technology for realizing "dual-drive" sampling. This comprehensive utility function integrates the prediction variance (statistical uncertainty index) from the Kriging module and the physical equation residuals (physical violation index) from the PINN module, using a weighted geometric average. The first index (Kriging prediction variance) quantifies the uncertainty of the Kriging model's scalar intensity prediction at the unsampled design point ω. This formula originates from Gaussian process regression theory: the larger the variance, the sparser the data near that point, and the less confident Kriging's prediction at that point. Therefore, maximizing this index tends to explore regions with sparse data. The second index (PINN physical residual) quantifies the degree of violation of the mechanical equilibrium equations and constitutive relations by the PINN-predicted full-field stress-strain distribution at the design point ω; specifically, it calculates the N values ​​randomly sampled throughout the entire structural geometric domain. c At each configuration point, the weighted average of the equilibrium equation residuals and constitutive residuals, and the mean residual value at the end of the initial training. and Normalization is performed. The larger the index, the more "physically contradictory" the stress field predicted by PINN is under this design variable, necessitating the supplementation of real simulation data at that point to correct the model. A weighted geometric mean is used to fuse the two normalized indices, rather than a simple weighted arithmetic mean, because the geometric mean has a "multiplicative" property: U is only large when both indices are not too small; if either index is close to 0, U will also approach 0. This forces the selected sampling points to contribute to both statistical uncertainty and physical contradictions, avoiding unilaterally exploring unknown regions while ignoring physical contradictions, or only correcting physical contradictions while ignoring unknown regions. The comprehensive utility function uses a weighted geometric mean to fuse the Kriging prediction variance and the PINN physical residuals. Variance represents cognitive uncertainty (to be explored), and physical residuals represent prediction errors (to be corrected). After fusion, the most valuable supplementary nodes can be accurately located. This solves the problems of blind, inefficient, and uncertain sampling that prioritizes uncertainty over physical accuracy in traditional methods, maximizing model quality with each supplementary simulation.

[0055] In an optional embodiment, the normalized first index and the normalized second indicator The results are obtained using the following formulas: , ,in, and These represent the maximum and minimum values ​​of the first index among multiple candidate design point sets. and These are the maximum and minimum values ​​of the second index in a set of multiple candidate design points, respectively, where δ is a preset positive constant to prevent the denominator from being zero.

[0056] In this embodiment, by normalizing the first and second indicators, the calculation of the comprehensive utility function is avoided due to differences in data range, the accuracy of selecting supplementary sample points is improved, and thus it is beneficial to build a more efficient and accurate target hybrid proxy model.

[0057] By using the minimax normalization method, the variance and residuals are scaled to the same order of magnitude, preventing one from overwhelming the other due to differences in magnitude. A small constant δ is introduced to prevent the denominator from being zero; δ is a very small positive number (e.g., 10). -n (n>3) to ensure computational stability; normalize the two indicators to the same [0,1] scale for comparability. Normalization is the basis for integrating multiple indicators, ensuring that the utility function truly reflects the node value. It solves the problem of different indicators having different dimensions and being unable to be directly integrated, ensuring that adaptive sampling is fair, reasonable, and scientific.

[0058] In an optional embodiment, the balancing weight coefficients are dynamically adjusted in each iteration as follows: ,in, Let Variance be the variance of the first index across multiple candidate design point sets. The variance of the second index over a set of multiple candidate design points.

[0059] In this embodiment, by dynamically adjusting the balancing weight coefficients according to a specific formula in each iteration, the proportions of the first indicator (the variance predicted by the Kriging module) and the second indicator (the residual of the physical equation predicted by the physical information neural network module) in the comprehensive utility function can be adaptively balanced based on the variance of the first and second indicators on multiple candidate design point sets. This allows the comprehensive utility function to more reasonably reflect the value of the candidate design points, thereby more accurately selecting supplementary sample points, improving the efficiency of sample utilization during model construction, and helping to construct a structural static strength surrogate model that combines high accuracy and physical consistency.

[0060] This embodiment provides an adaptive adjustment method for the balance weight coefficient α2 in the comprehensive utility function, enabling the sampling strategy to automatically adjust the emphasis of exploration and correction based on the spatial distribution characteristics of the two indicators in the current model. The distribution characteristics of the Kriging prediction variance and the PINN physical residual will change at different stages of adaptive sampling. In the initial stage, due to the sparse sample, the Kriging variance is generally large throughout the design space, and the variance differences between different candidate points may be small; simultaneously, after initial training, the PINN physical residual may be concentrated in certain local regions. At this time, using a fixed α2 (e.g., 0.5) may not fully utilize the most valuable information. The dynamic adjustment strategy adopted in this embodiment is: calculate the normalized variance of the two indicators on the current candidate point set, and then set α2 to... The ratio of the variances of the two variances to the sum of their variances. Its physical meaning is: if If the variance is large (indicating significant differences in the degree of physical contradiction among different candidate points), then α2 tends to be large, making U... The increase in the exponent α2 indirectly leads to The exponent (1-α²) decreases. When When the variation is large, it indicates that the indicator is already able to distinguish candidate points well, so its weight (1-α2) should be reduced to allow for smaller variations. This involves giving them greater weight, thereby balancing their contributions. This is a balancing strategy to ensure that both have appropriate influence in the integration process.

[0061] In an optional embodiment, the convergence condition includes at least one of the following: the number of iterations reaches a preset upper limit; the maximum value of the comprehensive utility function is less than a first preset threshold; the change in the data fitting loss of the physical information neural network module within a preset number of consecutive iterations is less than a preset loss threshold, and the residual of the physical equation corresponding to the physical information neural network module is less than a preset residual threshold.

[0062] In this embodiment, a variety of selectable convergence conditions are provided to terminate the iteration operation. When the number of iterations reaches a preset upper limit, infinite iteration can be avoided, saving computational resources. When the maximum value of the comprehensive utility function is less than the first preset threshold, it indicates that the benefits brought by supplementing samples are limited, and stopping the iteration can improve efficiency. When the data fitting loss of the physical information neural network module changes little and the residual of the physical equation is also small, it indicates that the model has reached a certain accuracy. At this time, stopping the iteration can ensure that the obtained target hybrid proxy model has good performance.

[0063] This embodiment provides convergence criteria for the iterative update step, i.e., when adaptive sampling can be stopped and the final target hybrid surrogate model can be output. This is crucial to ensuring the practicality and efficiency of the method, avoiding infinite iteration or premature termination. Specifically, this scheme provides three independent convergence criteria, and iteration can be stopped if any one of them is met, forming a multi-level judgment mechanism. The first is a condition based on computing resources: setting a maximum number of iterations (e.g., 20 rounds, or other iterations), and forcibly stopping when the preset upper limit is reached. This ensures that even if the model has not yet converged ideally, the algorithm can terminate within a controllable time, avoiding the risk of infinite loops due to some faults. The second is a condition based on sampling benefit: stopping when the maximum value of the comprehensive utility function is less than a first preset threshold; the comprehensive utility function represents the value of the "most valuable point" in the current candidate point set. If the maximum value is already very close to zero, such as 0.01 (or other values), it means that even the most valuable point in the design space has negligible information gain, and continuing sampling is meaningless, indicating that the model has fully converged. The third type is based on model accuracy: Two sub-conditions must be met simultaneously: the data fitting loss of PINN changes minimally (indicating that the model parameters have stabilized and further training will not significantly alter the prediction results), and the residual of the physical equation is less than a preset threshold (indicating that the model's predictions already conform to physical laws). This directly measures the quality of the model itself and is the most fundamental convergence criterion. This embodiment provides a multi-level convergence judgment mechanism, balancing efficiency, resource control, and model accuracy; users can choose the appropriate convergence conditions according to their actual needs. It solves the problems of single, unreliable convergence criteria and stopping before the model is fully trained, ensuring the model reaches an engineering-ready state.

[0064] In an optional embodiment, the Kriging module employs a general Kriging model, the regression basis function of which includes the scalar intensity response predicted by the physical information neural network module as an auxiliary variable; the general Kriging model takes the following form: Where v is the current design variable vector, y(v) is the scalar intensity response prediction value of the scheme corresponding to the current design variable v by the Kriging module, β0 is the constant term, and β1 is the regression coefficient. Z(v) represents the maximum value of the full-field stress-strain distribution predicted by the physical information neural network module for the current design variable v, and Z(v) represents the Gaussian process component.

[0065] In this embodiment, the Kriging module adopts a general Kriging model, and the regression basis function includes the scalar intensity response predicted by the physical information neural network module as an auxiliary variable, which can improve the prediction accuracy of the Kriging module for the scalar intensity response. Combined with the prediction results of the physical information neural network module, the prediction of the Kriging module is more accurate and reliable, and better adapts to the needs of actual engineering applications.

[0066] In this embodiment, the Kriging module employs a general Kriging model and cleverly incorporates PINN's prediction results as auxiliary information. Its regression basis function (trend term) includes the scalar intensity response predicted by PINN as an auxiliary variable. This achieves information sharing between PINN and Kriging, embedding prior physical knowledge into the statistical model. Conventional Kriging models interpolate based on only a small number of high-fidelity data points, while PINN can quickly provide a low-cost but physically consistent prediction for any design point. This embodiment uses the scalar intensity response predicted by PINN (such as maximum stress) as a regression basis function of the general Kriging model. This solves the technical problem that the Kriging model, relying entirely on statistical assumptions in sparse data regions, may fail to capture complex nonlinear trends. By introducing PINN's prediction as a "physical trend term," the Kriging model no longer guesses "from scratch," but is refined using high-fidelity data based on a reasonable trend provided by PINN. This greatly improves the prediction accuracy and generalization ability of the Kriging model, enabling the two parallel modules to truly achieve information fusion and performance complementarity. By introducing the physical trend information provided by PINN, the Kriging model can better capture global response behavior and make more accurate predictions even in regions far from the training samples. This allows PINN to not only provide physical residual metrics for active learning but also to guide the learning of the Kriging model, achieving a 1+1>2 effect. The introduction of PINN is equivalent to providing Kriging with a massive amount of low-fidelity data, which can further reduce the Kriging model's need for expensive high-fidelity data or achieve higher accuracy with the same amount of data.

[0067] In an optional embodiment, the above method further includes the application step: using the target hybrid surrogate model for static strength assessment at the new design point, outputting the predicted full-field stress-strain distribution corresponding to the new design point through the physical information neural network module, and outputting the predicted scalar strength response and the confidence interval of the predicted scalar strength response corresponding to the new design point through the Kriging module.

[0068] In this embodiment, the target hybrid surrogate model can be applied to the static strength assessment of the new design point. The physical information neural network module outputs the predicted full-field stress and strain distribution corresponding to the new design point, and the Kriging module outputs the predicted scalar strength response and the confidence interval of the predicted scalar strength response corresponding to the new design point, thereby achieving efficient assessment of the static strength of the structure.

[0069] The trained target hybrid surrogate model is applied to practical engineering applications—rapid static strength assessment. The user inputs a new design point, and the model outputs two results within milliseconds: the PINN module outputs a stress-strain distribution cloud map of the entire field (used to analyze structural weak points, failure modes, etc.), and the Kriging module outputs key scalar strengths (such as maximum stress and safety factor) and their confidence intervals (used for safety verification and reliability analysis). This achieves integrated "rapid full-field prediction + uncertainty assessment," replacing the time-consuming finite element method. It solves the problems of slow assessment speed, inability to perform batch calculations, and lack of uncertainty indication in engineering, supporting rapid design and safety assessment.

[0070] Optionally, the above method also includes: performing Monte Carlo simulations using a target hybrid surrogate model, calculating the failure probability based on the predicted variance provided by the Kriging module, and guiding the reliability optimization of design parameters. Using a target hybrid surrogate model for Monte Carlo simulations and calculating the failure probability based on the predicted variance provided by the Kriging module can provide a basis for the reliability optimization of design parameters, thereby guiding the reliability optimization of design parameters and improving the reliability and accuracy of structural strength analysis. By replacing the time-consuming FEM with a surrogate model, each sampling evaluation in the Monte Carlo simulation can be completed instantly. The predicted variance provided by the Kriging module represents the uncertainty of the surrogate model itself and can be included in the calculation of the failure probability, making the final result more rigorous. Combining the surrogate model with probabilistic analysis solves the problem that traditional reliability analysis requires massive simulations and is computationally infeasible, achieving efficient and high-precision reliability assessment. This makes efficient reliability optimization design possible, greatly improving the safety and robustness of products.

[0071] The following description, in conjunction with specific embodiments, illustrates a method for constructing a hybrid surrogate model for structural static strength that integrates physical information neural networks (PINNs) and Kriging. This method primarily addresses the low computational efficiency issues arising from numerous design variables and the high cost of high-precision finite element simulations in the static strength analysis of structural components. The aim is to construct a high-precision, high-efficiency hybrid surrogate model by integrating physical mechanism constraints with data-driven uncertainty quantification, thereby enabling rapid evaluation of the static strength of complex structures and reliable optimization design. PINN embeddings physical laws to ensure the physical consistency of field quantity predictions, Kriging models efficiently capture scalar responses and quantify uncertainties, and an adaptive point-addition criterion intelligently selects samples, thus achieving the construction of a high-precision, high-confidence hybrid surrogate model with minimal simulation samples.

[0072] This proposal suggests a hybrid surrogate model construction method integrating a physical information neural network (PIN) and Kriging for structural static strength. Its core improvement lies in constructing a hybrid architecture and adaptive training mechanism that combines "mechanism-data complementarity and global-local integration." The model consists of parallel PINN and Kriging layers. PINN embeds the static equilibrium equations and stress-strain constitutive relations as constraints into the network, responsible for predicting the stress / strain field distribution of the structure under limited samples, ensuring physical consistency. The Kriging model is responsible for predicting key scalar responses such as maximum equivalent stress and safety factor, as well as their uncertainties. Simultaneously, an adaptive point-addition criterion is designed, comprehensively considering the prediction variance of the surrogate model and the residuals of the PINN physical equations. This intelligently selects the most effective supplementary sample points for high-precision simulation, thereby gradually improving the global prediction capability of the hybrid surrogate model with the fewest simulations. Specifically, this may include the following steps: Step 1: Initialization of the Hybrid Agent Model Architecture. A hybrid architecture consisting of a Physical Information Neural Network (PINN) and a Kriging model connected in parallel is constructed. PINN receives design variables (such as geometric dimensions and thickness) and spatial coordinates as input nodes, outputting the stress / strain components at those coordinate points. Its loss function is weighted by a data fitting term (error between predicted and sample values) and a physical law term (equilibrium equation residuals, constitutive relation residuals), ensuring the network follows the laws of mechanics during training. The Kriging model receives the same design variables as input and outputs global performance scalars (such as maximum equivalent stress, displacement at a specific point). Its Gaussian process characteristics provide the mean and variance (uncertainty measure) of the predicted values. Both models share an initial small set of high-precision finite element samples (e.g., 20-50 samples) and undergo independent initialization training. PINN learns the physical field distribution, while Kriging learns the scalar response surface.

[0073] 1. Mathematical form and parameter interpretation of the loss function In the hybrid surrogate model, the Physical Information Neural Network (PINN) is responsible for predicting the field distribution, and its loss function is the core element ensuring physical consistency. The following composite loss function is recommended: ; The specific definitions of each item are as follows: (1) Data fitting loss term

[0074] This section measures the deviation between the network predictions and the finite element high-fidelity simulation samples (nodal stress / displacement): , (2) Physical law constraint loss term

[0075] This part forces the network output to satisfy the elasticity control equations by sampling collocation points within the computational domain: , The meanings of the parameters are explained in Table 1.

[0076] Table 1

[0077] Improvement 1: PINN's Adaptive Loss Balancing strategy.

[0078] Problem Background: Given a very small initial sample size (20-50), how does PINN perform? This term is highly prone to overfitting, and The sparse configuration points lead to insufficient gradient contribution, making the model prone to getting trapped in local optima where "the data fits very well but the physics is absurd".

[0079] Improvement solution: Adopt an adaptive weighting algorithm based on gradient statistics (such as Learning Rate Annealing or Neural Tangent Kernel weighting).

[0080] Specific steps: Every K iterations of training, calculate the ratio of the gradient norm of the data loss to the physical loss relative to the shared parameters of the last layer. .

[0081] Dynamic adjustment: setting λ D =1.0, λ P=γ·α1, where α1 is the initial physical weight coefficient (10-100 is recommended). This ensures that the physical constraints maintain a dynamic balance with the data terms in the early stages of training, thereby significantly improving the generalization smoothness of PINN's predicted stress field under small sample conditions.

[0082] Improvement point 2: Physics-Informed Trend Function for Kriging regression trend term.

[0083] Background: Traditional Kriging assumes that the mean function is an unknown constant. When dealing with the strong nonlinear variation of the maximum equivalent stress of a structure with design variables, a large number of samples are needed to construct an accurate variation function.

[0084] Improvement: Use the low-precision physical approximation solution as the explicit regression basis function (TrendFunction) of Kriging.

[0085] Specific operation: Utilizing another advantage of this scheme—PINN, although its accuracy is limited after initial training, its physical field trend is correct—the maximum stress prediction value of PINN at the initial sample points is taken. As auxiliary variables, a universal Kriging model is constructed in the following form: , Effect: This operation linearizes the structural mechanics trends captured by PINN (such as the quadratic relationship between the bending moment and length at the root of a cantilever beam) into the Kriging mean structure, making the prediction variance estimate of Kriging under the initial sparse samples more reasonable, and providing more reliable uncertainty guidance for the subsequent adaptive point addition criterion.

[0086] Improvement point 3: Feature engineering based on dual-model joint initialization of mutual information.

[0087] Background: Directly concatenating design variables (such as L, t) with spatial coordinates (x, y, z) into PINN can cause the network to learn spurious non-physical correlations, increasing the difficulty of training.

[0088] Improvement solution: Introduce Fourier feature mapping and decouple normalization of design variables.

[0089] PINN input layer processing: Applying Gaussian random Fourier feature mapping to spatial coordinates x. This is to alleviate the "spectral bias" problem in neural networks when fitting high-frequency stress concentration regions.

[0090] Dual-model interface alignment: When constructing the correlation matrix in Kriging, the kernel function distance metric no longer uses the original Euclidean design variable space, but instead uses the distance between the latent feature vectors output by the PINN encoder layers (the first few unfrozen layers). This helps to achieve a "soft alignment" of the two models in the manifold space, facilitating the simultaneous evaluation of field residuals and global variance by the adaptive point addition criterion in subsequent steps.

[0091] Step 2: Adaptive Sample Augmentation Based on Physics and Statistics. Based on the initial training results, an adaptive sampling point addition criterion is implemented to optimize the sample space. The system comprehensively evaluates two indicators to select new sampling points: first, based on the prediction variance of the Kriging model, it identifies the design region with the greatest uncertainty in scalar response prediction to explore the unknown space; second, based on the physical equation residuals predicted by PINN, it identifies regions where the equilibrium equations are not satisfied to a large extent or where physical laws are severely violated to correct physical inconsistencies. These two indicators are integrated into a comprehensive utility function, and the design point with the largest utility value is selected as the supplementary sample point for a new round of high-precision finite element analysis. The newly obtained data is added to the training set, and the hybrid model parameters are iteratively updated.

[0092] 1. Definition and Quantitative Expression of Two Core Indicators (1) Kriging prediction variance index

[0093] Kriging model at any unsampled design point Provides the posterior variance of the Gaussian process. This variance characterizes the uncertainty of the statistical model's prediction of the global scalar response (such as the maximum equivalent stress). , Where, σ Z ² represents the process variance, R is the correlation matrix, and r is the correlation vector between the test point and the sample points.

[0094] Physical meaning: The larger this value is, the more likely the design region has not been fully covered by the samples. Exploring this point can help improve the statistical accuracy of the global proxy model.

[0095] (2) PINN physical equation residual index

[0096] For a given design variable, PINN can predict the stress field throughout the entire structural domain Ω. The normalized average physical residual is defined as a measure of the degree to which the design point violates the laws of mechanics: , in and These are the mean values ​​of the equilibrium equation residuals and constitutive residuals of PINN on the training set after initial training, respectively, and are used as the normalization benchmark.

[0097] Physical meaning: The larger the value, the more likely PINN is to fail to fully satisfy the mechanical control equations in the structural field prediction corresponding to the design variable, and high-fidelity data for this point is urgently needed to correct local physical deviations.

[0098] 2. Mathematical Form and Fusion Strategy of the Comprehensive Utility Function Fusion objective: To achieve a balance between exploring regions with high statistical uncertainty and correcting regions with severe physical violations. It is recommended to use a weighted geometric average based on the normalization exponent to ensure that neither index is negligible. , in: The normalized value of the Kriging variance index: , Here is the normalized value of the PINN residual index: , α2ϵ[0,1] is the balancing weight coefficient, which controls the emphasis of exploration and correction.

[0099] , These represent the maximum and minimum variances within the current candidate point set; and Similarly, δ is a very small positive value to prevent the denominator from being zero.

[0100] An adaptive method for determining the balancing weight coefficient α2: In each round of adaptive iteration, the variance ratio of the two normalized indices on the candidate point set is calculated to dynamically adjust α2.

[0101] , Principle: If the PINN residual index has large spatial variation (uneven distribution of physical violation regions), the complement weight of α2 is automatically increased to prioritize the correction of physical errors; conversely, if the Kriging variance distribution is relatively discrete, it tends to explore statistical blind spots.

[0102] Step 3: Rapid Assessment and Optimization of Structural Static Strength. After model training convergence, a hybrid surrogate model is used for rapid assessment. For any new design point, the system outputs two results in milliseconds: one is the full-field stress-strain distribution predicted by PINN, which, due to its embedded physical constraints, ensures the rationality and physical consistency of the distribution; the other is the extreme stress and safety factor predicted by Kriging, along with the corresponding confidence intervals. In reliability optimization design scenarios, this hybrid model is directly called to perform tens of thousands of Monte Carlo simulations. Kriging's uncertainty quantification is used to quickly assess the failure probability, guiding iterative optimization of design parameters, thereby significantly reducing computational costs while maintaining accuracy.

[0103] This application's embodiments overcome the shortcomings of pure data-driven proxy models, such as poor generalization ability and potential physical fallacies in small samples, by integrating physical mechanisms and data-driven approaches. At the same time, it utilizes the uncertainty quantification capability of the Kriging model to guide adaptive sampling, realizing an efficient design paradigm of "gaining big knowledge from small samples." This method makes it possible to perform millisecond-level static strength assessment and large-scale reliability optimization design for complex structural parts while ensuring computational accuracy, significantly shortening the R&D cycle.

[0104] This application also provides a surrogate model construction apparatus for structural strength analysis, used to execute the surrogate model construction method for structural strength analysis in any of the foregoing embodiments, such as... Figure 2 As shown, Figure 2 This is a structural block diagram of a surrogate model construction device for structural strength analysis provided in an embodiment of this application. The device includes: The acquisition module is used to acquire the initial training set, which contains finite element simulation results corresponding to multiple design samples. The finite element simulation results include full-field stress-strain distribution data and scalar strength response data. The module is used to construct a parallel physical information neural network module and a Kriging module, and to train the physical information neural network module and the Kriging module respectively using an initial training set. The physical information neural network module is used to output the predicted full-field stress-strain distribution based on the input design variables and spatial coordinates. The loss function of the physical information neural network module includes a data fitting loss term and a physical constraint loss term. The Kriging module is used to output the predicted scalar strength response based on the input design variables and provide the prediction variance. The iteration module performs the following iterative operations until a preset convergence condition is met to obtain the target hybrid surrogate model: Multiple candidate design points are selected from the unsampled design points in the design space; the comprehensive utility function value of the multiple candidate design points is calculated, where the candidate design points are different from the sampled design points in the initial training; the comprehensive utility function is obtained by fusing the first index and the second index; the first index is the prediction variance of the Kriging module for the multiple candidate design points; the second index is the physical equation residual obtained from the full-field stress-strain distribution calculation of the multiple candidate design points based on the prediction of the physical information neural network module; the candidate design point with the largest comprehensive utility function value is selected as a supplementary sample point; finite element simulation is performed on the supplementary sample point to obtain supplementary data; the supplementary data is added to the initial training set to update the training set; and the physical information neural network module and the Kriging module are retrained based on the updated training set; it is determined whether the convergence condition is met; if it is met, the iteration operation is stopped; otherwise, the next round of iteration operation is continued.

[0105] It should be noted that the devices or systems provided in the above embodiments are only illustrated by the division of the above functional modules. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the device can be divided into different functional modules to complete all or part of the functions described above. In addition, the system and method embodiments provided in the above embodiments belong to the same concept. Other device or system embodiments correspond to the aforementioned method embodiments. Other technical features are described in the previous embodiments and will not be repeated here.

[0106] This application also provides a computer-readable storage medium storing instructions that, when executed, perform the steps of any of the methods described above.

[0107] In one exemplary embodiment, the aforementioned computer-readable storage medium may include, but is not limited to, various media capable of storing computer programs, such as a USB flash drive, read-only memory (ROM), random access memory (RAM), portable hard disk, magnetic disk, or optical disk.

[0108] This application also discloses an electronic device. For example... Figure 3 As shown, Figure 3 This is a schematic diagram of the structure of an electronic device disclosed in an embodiment of this application. The electronic device 300 may include: at least one processor 301, at least one network interface 304, a user interface 303, a memory 305, and at least one communication bus 302.

[0109] The communication bus 302 is used to enable communication between these components.

[0110] The user interface 303 may include a display screen and a camera. Optionally, the user interface 303 may also include a standard wired interface and a wireless interface.

[0111] The network interface 304 may optionally include a standard wired interface or a wireless interface (such as a Wi-Fi interface).

[0112] The processor 301 may include one or more processing cores. The processor 301 connects to various parts of the electronic device (such as a server) using various interfaces and lines, and performs various server functions and processes data by running or executing instructions, programs, code sets, or instruction sets stored in memory 305, and by calling data stored in memory 305. Optionally, the processor 301 may be implemented using at least one hardware form of Digital Signal Processing (DSP), Field-Programmable Gate Array (FPGA), or Programmable Logic Array (PLA). The processor 301 may integrate one or a combination of several of the following: Central Processing Unit (CPU), Graphics Processing Unit (GPU), and modem. The CPU primarily handles the operating system, user interface, and applications; the GPU is responsible for rendering and drawing the content required for display; and the modem handles wireless communication. It is understood that the modem may also not be integrated into the processor 301 and may be implemented as a separate chip.

[0113] The memory 305 may include random access memory (RAM) or read-only memory. Optionally, the memory 305 may include a non-transitory computer-readable storage medium. The memory 305 may be used to store instructions, programs, code, code sets, or instruction sets. The memory 305 may include a program storage area and a data storage area, wherein the program storage area may store instructions for implementing an operating system, instructions for at least one function (such as touch function, sound playback function, image playback function, etc.), instructions for implementing the above-described method embodiments, etc.; the data storage area may store data involved in the above-described method embodiments, etc. Optionally, the memory 305 may also be at least one storage device located remotely from the aforementioned processor 301. (Refer to...) Figure 3 The memory 305, which serves as a computer storage medium, may include an operating system, a network communication module, a user interface module, and an application program for constructing a proxy model for structural strength analysis.

[0114] exist Figure 3 In the illustrated electronic device 300, the user interface 303 is mainly used to provide an input interface for the user and to acquire user input data; while the processor 301 can be used to call an application program for constructing a proxy model for structural strength analysis stored in the memory 305. When executed by one or more processors 301, the electronic device 300 performs one or more of the methods described in the above embodiments. It should be noted that, for the foregoing method embodiments, for the sake of simplicity, they are all described as a series of actions. However, those skilled in the art should understand that this application is not limited to the described order of actions, because according to this application, some steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also understand that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily essential to this application.

[0115] In the above embodiments, the descriptions of each embodiment have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0116] In the various embodiments provided in this application, it should be understood that the disclosed apparatus or system can be implemented in other ways. For example, the apparatus or system embodiments described above are merely illustrative; for instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some service interface; the indirect coupling or communication connection between apparatuses or units may be electrical or other forms.

[0117] The above description is merely an exemplary embodiment of this disclosure and should not be construed as limiting the scope of this disclosure. Any equivalent changes and modifications made in accordance with the teachings of this disclosure shall still fall within the scope of this disclosure. Other embodiments of this disclosure will be readily apparent to those skilled in the art upon consideration of the disclosure herein.

[0118] This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art that are not described in this disclosure.

Claims

1. A method for constructing a surrogate model for structural strength analysis, characterized in that, include: Obtain an initial training set, which contains finite element simulation results corresponding to multiple design samples, wherein the finite element simulation results include full-field stress-strain distribution data and scalar strength response data; A parallel physical information neural network module and a Kriging module are constructed, and the physical information neural network module and the Kriging module are trained respectively using the initial training set. The physical information neural network module is used to output the predicted full-field stress-strain distribution based on the input design variables and spatial coordinates. The loss function of the physical information neural network module includes a data fitting loss term and a physical constraint loss term. The Kriging module is used to output the predicted scalar strength response based on the input design variables and provide the prediction variance. Perform the following iterative operations until the preset convergence condition is met to obtain the target hybrid surrogate model: Select multiple candidate design points from the unsampled design points in the design space, calculate the comprehensive utility function value of the multiple candidate design points, wherein the candidate design points are different from the design points sampled in the initial training, and the comprehensive utility function is obtained by fusing a first index and a second index; the first index is the prediction variance of the Kriging module for the multiple candidate design points; the second index is the physical equation residual calculated based on the full-field stress-strain distribution of the multiple candidate design points predicted by the physical information neural network module; select the candidate design point with the largest comprehensive utility function value as a supplementary sample point; perform finite element simulation on the supplementary sample point to obtain supplementary data, add the supplementary data to the initial training set to update the training set, and retrain the physical information neural network module and the Kriging module based on the updated training set; determine whether the convergence condition is met, if it is met, stop the iterative operation, if it is not met, continue to perform the next round of iterative operation.

2. The method according to claim 1, characterized in that, The loss function of the physical information neural network module takes the following form: ； in, Let θ be the loss function of the physical information neural network module, and let θ be the set of network parameters of the physical information neural network module. The loss term is used to fit the data. Let λ be the physical constraint loss term. D λ represents the weighting coefficient of the data fitting loss term. P This represents the weighting coefficient of the physical constraint loss term.

3. The method according to claim 2, characterized in that, The data fitting loss term The data fitting loss term is used to measure the deviation between the predicted value and the corresponding point in the finite element simulation result. Adopt the following form: ， Where, N d x is the number of sample points used for supervised training. i Let v be the spatial coordinates of the i-th sampling point. i Let i be the design variable vector corresponding to the i-th sampling point. The displacement field predicted by the physical information neural network module, u i The displacement reference value corresponding to the i-th sampling point is obtained from the finite element simulation. The stress field predicted by the physical information neural network module, σ i The stress reference value corresponding to the i-th sampling point is obtained from the finite element simulation. The physical constraint loss term The physical constraint loss term is used to force the prediction results to satisfy the elasticity control equations. Adopt the following form: ， Where, N c x represents the number of randomly sampled physical constraint placement points within the structural geometry domain. j Let v be the spatial coordinate of the j-th sampling point in the randomly sampled physical constraint configuration points. j Let b be the design variable vector corresponding to the j-th sampling point in the randomly sampled physical constraint configuration points, where b is the physical force term. Let ε be the fourth-order elastic stiffness tensor of the material, and ε be the displacement based on the predicted displacement. The strain tensor is obtained by calculating the geometric equation.

4. The method according to claim 2, characterized in that, The loss function of the physical information neural network module adopts an adaptive weighting strategy, dynamically adjusting the weight coefficients of the data fitting loss term and the physical constraint loss term during training.

5. The method according to claim 4, characterized in that, The adaptive weighting strategy includes: During the training process of the physical information neural network module, after a preset number of training cycles, the ratio of the gradient norm of the data fitting loss to the gradient norm of the physical constraint loss is calculated. and according to λ P =γ·α1 dynamically adjusts the weight coefficient λ of the physical constraint loss term. P , and λ D =1, where α1 is the preset initial physical weight coefficient.

6. The method according to claim 1, characterized in that, The comprehensive utility function combines the first indicator and the second indicator in the following form: ， in, Design points that were not sampled The first indicator after normalization. α2 is the normalized second index, and α2 is the balance weight coefficient, α2∈[0,1]; First indicator Adopt the following form: ， Where, σ Z ² represents the process variance, R is the correlation matrix of the Kriging model, r is the correlation vector between the design point to be tested and the existing sample points, and 1 is the unit column vector; Second indicator Adopt the following form: ， Where, N c x represents the number of physical constraint placement points randomly sampled within the structural geometry domain. k Let θ be the spatial coordinates of the k-th configuration point, and θ be the set of network parameters of the physical information neural network module. This represents the mean of the equilibrium equation residuals of the physical information neural network module on the training set after initial training. This represents the mean constitutive relation residual of the physical information neural network module on the training set after initial training.

7. The method according to claim 6, characterized in that, The normalized first index and the normalized second index The results are obtained using the following formulas: ， ， in, and These represent the maximum and minimum values ​​of the first index among multiple candidate design point sets. and These are the maximum and minimum values ​​of the second index in a set of multiple candidate design points, respectively, where δ is a preset positive constant to prevent the denominator from being zero.

8. The method according to claim 6, characterized in that, The balancing weight coefficients are dynamically adjusted in each iteration as follows: ， in, Let Variance be the variance of the first index across multiple candidate design point sets. The variance of the second index over a set of multiple candidate design points.

9. The method according to claim 1, characterized in that, The convergence condition includes at least one of the following: The number of times the iterative operation is executed reaches a preset upper limit; The maximum value of the comprehensive utility function is less than a first preset threshold; The change in the data fitting loss of the physical information neural network module within a preset number of consecutive iterations is less than a preset loss threshold, and the residual of the physical equation corresponding to the physical information neural network module is less than a preset residual threshold.

10. The method according to claim 1, characterized in that, The Kriging module adopts a general Kriging model, and the regression basis function of the general Kriging model includes the scalar intensity response predicted by the physical information neural network module as an auxiliary variable. The general Kriging model takes the following form: ， Where v is the current design variable vector, y(v) is the scalar intensity response prediction value of the scheme corresponding to the current design variable v by the Kriging module, β0 is the constant term, and β1 is the regression coefficient. Z(v) represents the maximum value of the full-field stress-strain distribution predicted by the physical information neural network module for the current design variable v, where Z(v) is the Gaussian process component.

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