A method for constructing a proxy model for high-dimensional CAE simulation
By constructing a surrogate model through linear combination of multiple types of basis functions and a two-stage optimization strategy, the problems of computational complexity and training efficiency in high-dimensional nonlinear CAE simulation are solved, and efficient and stable prediction and modeling accuracy are improved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- PERA
- Filing Date
- 2026-02-12
- Publication Date
- 2026-06-05
Smart Images
Figure CN122154427A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of CAE simulation technology, and in particular to a method for constructing a proxy model for high-dimensional CAE simulation. Background Technology
[0002] Computer-aided engineering (CAE) simulation technology is a core tool for modern industrial design and analysis, widely used in aerospace, automotive manufacturing, energy equipment, and electronics, among other fields, to predict the performance, reliability, and safety of products before physical manufacturing. Traditional CAE simulation methods primarily rely on physical principles to establish mathematical models and solve partial differential equations using numerical calculations to predict product performance under different operating conditions. However, with the increasing complexity of engineering problems, the dimensionality of simulation model parameters is growing, and nonlinear response characteristics are becoming more pronounced, posing a significant challenge to traditional simulation technologies.
[0003] In high-dimensional parameter space modeling, traditional methods such as response surface methodology and Kriging interpolation can establish input-output relationships to some extent. However, as the parameter dimension increases, these methods face the "curse of dimensionality," with computational complexity growing exponentially, making it difficult to meet the real-time response requirements of engineering practice. In handling nonlinear problems, traditional methods often rely on expert experience to manually select basis function forms, lacking adaptability and easily getting trapped in local optima.
[0004] In recent years, with the rapid development of artificial intelligence technology, machine learning methods have shown great potential in the field of CAE simulation. Neural networks, support vector machines, and other machine learning methods have solved nonlinear modeling problems to some extent, but they still have limitations such as low training efficiency and insufficient generalization ability when dealing with high-dimensional data. Especially when training data is limited, these methods are prone to overfitting, affecting the engineering practicality of the model. Summary of the Invention
[0005] Based on the above analysis, the embodiments of the present invention aim to provide a proxy model construction method for high-dimensional CAE simulation, in order to solve the problems of inflexible modeling and low prediction accuracy in existing high-dimensional, nonlinear CAE simulation scenarios.
[0006] This invention provides a method for constructing a proxy model for high-dimensional CAE simulation, comprising the following steps:
[0007] A historical dataset is constructed based on the design parameter vector in the CAE simulation task and the physical response values obtained through CAE simulation calculation. A surrogate model is constructed by linearly combining multiple basis functions of different types; the internal parameter set of each basis function in the surrogate model is initialized based on the statistical characteristics of historical datasets. Using historical datasets, a two-stage optimization strategy is employed to train the surrogate model, resulting in a well-trained surrogate model used to predict the physical response values corresponding to the new design parameter vectors in CAE simulation tasks.
[0008] Based on further improvements to the above method, the basis functions include: linear functions, sigmoid activation functions, and Gaussian radial basis functions; the surrogate model is expressed by the following formula: , in, The proxy model represents the design parameter vector. Predicted physical response value, Indicates the global bias term. Indicates the first basis functions The linear weighting coefficients, Indicates the first basis functions The internal parameter set, This indicates the number of basis functions.
[0009] Based on further improvements to the above method, the type and number of basis functions are determined adaptively, including: By identifying the relationship patterns between design parameter vectors and physical response values in historical datasets, the type of basis function is adaptively selected. A validation set is defined from the historical dataset. Based on the initial number of each type of basis function, the final number of each type of basis function is adaptively determined by incrementally building and comparing the performance of the surrogate model before and after the increment on the validation set.
[0010] Based on the further improvement of the above method, the internal parameter set of the sigmoid activation function includes: a weight vector and local bias terms; the internal parameter set of the sigmoid activation function in the surrogate model is initialized, including: Calculate the covariance matrix of the design parameter vectors in the historical dataset and perform principal component analysis to obtain the previous... One principal component and its eigenvalues; Obtain the corresponding random coefficients based on the eigenvalues of each principal component, and then use each random coefficient to... The principal components are linearly combined to obtain the initial values of the weight vector for each sigmoid activation function; The initial values of the local bias terms for each sigmoid activation function are obtained based on the weight vector, the mean vector of the design parameter vector, and the quantile of the physical response value.
[0011] Based on further improvements to the above method, the internal parameter set of the Gaussian radial basis function includes: center point and bandwidth; the internal parameter set of the Gaussian radial basis function in the surrogate model is initialized, including: The number of Gaussian radial basis functions is used as the number of clusters, and a clustering algorithm is used to cluster the design parameter vectors in the training dataset; The vector of each cluster center point is used as the initial value of the center point of each Gaussian radial basis function; the initial value of the bandwidth of each Gaussian radial basis function is obtained based on the average distance from the design parameter vector within each cluster to its center point and the scaling factor.
[0012] Based on further improvements to the above method, a two-stage optimization strategy includes: In each iteration of the first stage, the learning rate is dynamically adjusted according to the adjacent gradient signs of each parameter, thereby obtaining the first update amount of each parameter and updating it; when the error reduction rate of multiple consecutive iterations is lower than a preset threshold, the first stage ends and the second stage begins. In each iteration of the second stage, based on the parameters and surrogate model optimized in the first stage, a linear equation for the Jacobian matrix with an adaptive damping factor is constructed and solved to obtain the second update amount for each parameter and update it; when the iteration termination condition is met, the final optimized parameters are obtained.
[0013] Based on the further improvement of the above method, the learning rate is dynamically adjusted according to the adjacent gradient signs of each parameter using the following formula: , in, and They represent the first Second and third In the nth iteration Parameters The learning rate; and These represent the preset maximum and minimum learning rates, respectively. and These represent the preset learning rate growth factor and decay factor, respectively. and They represent the first Second and third Loss function in the next iteration and For parameters The gradient; and These represent the minimum value function and the maximum value function, respectively.
[0014] Based on the further improvement of the above method, the adaptive damping factor is calculated by updating the loss function value of the surrogate model after the parameters are updated according to the second update amount obtained in the current iteration. If the loss function value decreases compared to before the update, the parameters are updated according to the second update amount in the current iteration, and the damping factor is reduced by a coefficient less than 1 before entering the next iteration; otherwise, the damping factor is increased by a coefficient greater than 1, and the second update amount in the current iteration is recalculated until the loss function value decreases compared to before the update.
[0015] Based on the further improvement of the above method, the corresponding random coefficients are obtained according to the eigenvalues of each principal component by random sampling from a normal distribution with a mean of zero and a variance equal to the reciprocal of the square root of the eigenvalues of that principal component.
[0016] Based on a further improvement of the above method, the initial value of the local bias term for each sigmoid activation function is obtained through the following formula: , in, Indicates the first Local bias terms of a sigmoid activation function Indicates the first The weight vector of each sigmoid activation function. The mean vector of the design parameter vector. Indicates the quantiles of the physical response value. This indicates the transpose operation.
[0017] Compared with the prior art, the present invention can achieve at least one of the following beneficial effects: 1. By constructing a surrogate model through linear combination of multiple types of basis functions, the surrogate model possesses the inherent ability to uniformly describe linear and nonlinear, global and local features, overcoming the limitation of the expressive power of traditional single-type basis function models. Data-driven initialization places parameters at a high-quality starting point, avoiding optimization from getting trapped in unfavorable local minima from the source. The two-stage optimization achieves a "stable first, then precise" convergence path. This makes the surrogate model training converge faster and the convergence process more stable and reliable, significantly reducing the dependence on hyperparameter tuning and the risk of training failure. It automatically constructs a surrogate model with powerful expressive power and efficient training characteristics for complex physical simulation problems, greatly improving prediction accuracy.
[0018] 2. By using an adaptive method to determine the type ratio and quantity of basis functions, the "self-organization" of the surrogate model structure is achieved. This not only improves the flexibility and accuracy of modeling, but also achieves the best balance between model complexity and generalization ability through performance comparison on the validation set.
[0019] 3. The first stage of the two-stage optimization strategy adopts an adaptive learning rate method that relies solely on the gradient sign, which is insensitive to noise and can stably and quickly enter the optimal region. The second stage adopts a higher-order optimization method with an adaptive damping factor, which dynamically adjusts the search strategy according to the error changes and converges quickly when approaching the optimal point. This two-stage collaborative approach enables the entire training process to cope with the non-convexity challenge of the initial stage and complete the final fine-tuning.
[0020] In this invention, the above-described technical solutions can be combined with each other to achieve more preferred combinations. Other features and advantages of this invention will be set forth in the following description, and some advantages may become apparent from the description or be learned by practicing the invention. The objects and other advantages of this invention can be realized and obtained from what is particularly pointed out in the description and drawings. Attached Figure Description
[0021] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts. Figure 1 This is a flowchart of a proxy model construction method for high-dimensional CAE simulation in an embodiment of the present invention. Detailed Implementation
[0022] Preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form part of this application and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.
[0023] A specific embodiment of the present invention discloses a method for constructing a proxy model for high-dimensional CAE simulation, such as... Figure 1 As shown, it includes steps S1-S3.
[0024] S1. Construct a historical dataset based on the design parameter vector in the CAE simulation task and the physical response values obtained through CAE simulation calculation.
[0025] It should be noted that when performing CAE simulation tasks, the design parameters (input variables) and physical responses (output variables) must first be defined. The design parameters include at least one of the following: geometrical parameters, material property parameters, load condition parameters, or boundary condition parameters; the physical responses include at least one of the following: structural mass properties, mechanical responses, thermal responses, fluid dynamic responses, or multidisciplinary coupled responses.
[0026] For example, geometric parameters include: thickness, length, diameter, and fillet radius of the component; material property parameters include: elastic modulus, Poisson's ratio, density, and thermal conductivity; load and boundary condition parameters include: magnitude, direction, pressure value, and constraint location of the load force; structural mass properties include: total mass of the vehicle body and total mass of the anti-collision beam; mechanical response includes: maximum equivalent stress, maximum deformation displacement, and natural frequency; thermal response includes: maximum temperature and average heat flux density; fluid dynamic response includes: drag coefficient and lift coefficient; and multidisciplinary coupling response includes: thermal stress under thermo-structural coupling.
[0027] Taking CAE simulation of car crashes as an example, six key design variables that significantly affect crash performance are selected to form a 6-dimensional input vector, including: bumper thickness (mm), front anti-collision beam material strength (MPa), energy absorption box length (mm), initial collision speed (km / h), material yield strength of key structural components (MPa), and structural stiffness coefficient. These parameters collectively define a design scheme to be evaluated. The simulation results focus on the maximum peak acceleration (unit: g) experienced by the passenger compartment, which is a core indicator for measuring crash safety. In practice, multiple responses (such as energy absorption and intrusion amount) can also be predicted in parallel, and the method in this embodiment is equally applicable.
[0028] Furthermore, multiple sample points are selected within a predefined design space (i.e., the range of variation of each parameter) using the Latin hypercube sampling method. For each sample point, a high-fidelity CAE simulation is performed to obtain its corresponding true physical response value.
[0029] Finally, all input-output pairs, i.e., the input design parameter vectors and the corresponding physical response values, are standardized and organized into a structured historical dataset, represented as follows: ,in, Indicates the first A vector of design parameters, Represents the dimension of the design parameter vector; express The corresponding physical response value; Indicates the amount of historical data.
[0030] Furthermore, the historical dataset is randomly divided into three parts: a training set (approximately 70%), a validation set (approximately 15%), and a test set (approximately 15%). The training set is used for model parameter optimization, the validation set for model structure selection and hyperparameter tuning, and the test set is used for final evaluation of the model's generalization performance, without participating in any adjustments during the entire modeling process. These are standard practices and will not be elaborated upon further.
[0031] S2. Construct a surrogate model by linearly combining multiple basis functions of different types; initialize the internal parameter set of each basis function in the surrogate model based on the statistical characteristics of the historical dataset.
[0032] It should be noted that this step is used to build a flexible and expressive surrogate model framework, and to use a data-driven strategy to provide high-quality initial values for the model parameters, thereby laying the foundation for subsequent efficient and stable optimization training.
[0033] The surrogate model in this embodiment approximates the complex mapping relationship from design parameters to physical response by linearly combining multiple different types of basis functions. These basis functions include linear functions, sigmoid activation functions, and Gaussian radial basis functions; each type of basis function represents a different feature learning capability, and the surrogate model includes at least two types of basis functions.
[0034] It should be noted that linear functions S-shaped activation function is used to capture approximately linear response trends in the design space. Gaussian radial basis functions are used to simulate nonlinear asymptotic behavior. They exhibit local response characteristics, effectively capturing highly localized nonlinear abrupt changes; the formulas for these three types of basis functions are shown below: , in, Represents a design parameter vector. This represents the coefficient vector in a linear function. and Let these represent the weight vector and local bias term in the sigmoid activation function, respectively. and Let represent the center point and bandwidth of the Gaussian radial basis function, respectively; Indicates the transpose operation; This indicates taking the L2 norm.
[0035] Furthermore, a surrogate model is constructed by linearly combining multiple basis functions of different types using the following formula: , in, The proxy model represents the design parameter vector. Predicted physical response value, This represents the global bias term, used to adjust the baseline output level of the model; Indicates the first basis functions The linear weighting coefficients represent the contribution weights of the basis function to the final prediction; Indicates the number of basis functions. Indicates the first basis functions The internal parameter set of , for linear functions, is the coefficient vector. For the sigmoid activation function, the weight vector is... and local bias terms For the Gaussian radial basis function with center point and bandwidth .
[0036] The surrogate model constructed in this embodiment integrates different types of basis functions, possessing the inherent ability to uniformly describe linear and nonlinear, global and local features, thus overcoming the limitation of the expressive power of traditional single-type basis function models.
[0037] Furthermore, the type of basis function is adaptively selected by identifying patterns in the relationship between the design parameter vector and the physical response values in the historical dataset, including: If the input design parameters and the output physical response values show a strong linear correlation, then a linear function should be preferred as the basis function. If the physical response value exhibits saturation or smooth transition characteristics as the design parameters change, then the sigmoid activation function should be preferred as the basis function. If the physical response value exhibits highly nonlinear or abrupt characteristics in a specific region of the design space, then the Gaussian radial basis function should be preferentially selected as the basis function.
[0038] It should be noted that the identification of the above relationship patterns can be achieved using judgment methods based on data statistical characteristics, including: identifying linear relationships by calculating the Pearson linear correlation coefficient, identifying monotonic nonlinearity by comparing the Spearman correlation coefficient, and identifying abrupt change characteristics by analyzing local variance, etc.
[0039] Specifically, the absolute value of the Pearson linear correlation coefficient between each input design parameter and the output response value is calculated. If at least one of these coefficients is greater than a set linear threshold, the linear function is selected as a candidate basis function. The absolute value of the Spearman correlation coefficient between each input design parameter and the output response value is also calculated. If it is significantly higher than the corresponding absolute value of the Pearson linear correlation coefficient, and the rate at which the output response value increases / decreases with the input design parameter (fitted through a moving window) significantly decreases and approaches zero, then saturation is indicated, and the sigmoid activation function is selected as a candidate basis function. The design space is divided into multiple local neighborhoods using K-nearest neighbors. If the ratio of the variance of the output response value within a local neighborhood to the variance of the global output response value is greater than a set Gaussian threshold, then the output within the local neighborhood changes drastically, and the Gaussian radial basis function is selected as a candidate basis function.
[0040] It is understood that the specific methods described above are merely examples. Any technical means capable of achieving similar relationship pattern recognition can be used to replace or in combination with the methods provided in this embodiment to complete adaptive type selection. This embodiment is not limited to the specific recognition algorithms listed.
[0041] The number of basis functions in the surrogate model is determined adaptively based on data and validation feedback. This includes: based on the initial number of each type of basis function, the final number of each type of basis function is determined adaptively by incremental construction and performance comparison of the surrogate model before and after the increment on the validation set.
[0042] Specifically, the initial construction involves building a minimal surrogate model, meaning that each selected candidate basis function is initially set to a small number, for example, starting with one basis function of each type. The surrogate model of linear combination is then trained in the first round using the training set to obtain the first relatively stable initial model. The calculator's error on the validation set .
[0043] From the initial model The process begins an iterative process, at the... During rounds of iteration: ① Determine the types of base functions to be added in this round, for example, try adding one base function for each type in a fixed order; ②In the model Based on this, add a base function of a selected type to form a candidate model. ; ③ Perform a limited number of iterations of optimization training on the candidate model. The purpose is to allow the newly added basis functions to initially adapt to the model, and then evaluate them on the validation set. error ; ④If Less than the error of the previous round If the relative decrease is greater than a preset threshold, then this round of additions is accepted. , Otherwise, reject this round of additions and revert the model to its previous state. The incremental construction process terminates when any termination condition is met; otherwise, the next iteration begins.
[0044] It should be noted that the termination conditions include: failure to successfully accept new basis functions in multiple consecutive iterations; or, the total number of basis functions in the surrogate model reaching a preset maximum value; or, the decrease in verification error after multiple successful additions being less than a preset minimum threshold.
[0045] The incremental construction method described above allows the surrogate model to obtain the most suitable internal structure based on the data characteristics of the specific CAE simulation task. This fundamentally solves the problems of insufficient modeling flexibility and susceptibility to underfitting or overfitting caused by the reliance on expert experience to manually select the form and number of basis functions in traditional methods. The introduction of a validation set ensures an optimal balance between model complexity and generalization ability.
[0046] For example, in a car crash simulation scenario, an adaptive process was used to determine a model containing eight basis functions: three Gaussian radial basis functions, three sigmoid activation functions, and two linear functions.
[0047] Among them, the three Gaussian radial basis functions are used to capture nonlinear abrupt changes or drastic variations in the maximum acceleration in specific local regions of the input space: The first Gaussian radial basis function focuses on "high-speed collision mutation", that is, when the combination of collision velocity and structural stiffness coefficient exceeds a certain critical threshold, the maximum acceleration increases nonlinearly and sharply; The second Gaussian radial basis function focuses on the "material-stiffness coupling sensitive region", that is, when the material yield strength is in a moderate range, but forms a specific combination with the strength of the anti-collision beam, the effect on the maximum acceleration exhibits a highly nonlinearity; The third Gaussian radial basis function focuses on the "structural geometry sensitive point", that is, when the ratio of the energy-absorbing box length to the bumper thickness is in a certain range, there is an optimal local peak in the suppression effect on the maximum acceleration.
[0048] Three sigmoid activation functions are used to simulate the saturation effect where the maximum acceleration gradually approaches a limit value as certain parameters increase: The first sigmoid activation function is used to simulate the "saturation of benefits from increased thickness", that is, as the thickness of the bumper increases, its effect on reducing maximum acceleration gradually weakens and tends to saturate; The second sigmoid activation function is used to simulate the "saturation of benefits from material strengthening", that is, as the strength of the crash beam or the yield strength of the material increases, its marginal benefit in reducing the maximum acceleration decreases.
[0049] The third sigmoid activation function is used to simulate the "acceleration plateau of low-speed collisions", that is, at low collision speeds (e.g., <30 km / h), the maximum acceleration itself changes slowly with speed and tends to a stable plateau.
[0050] Two linear functions are used to describe the approximate linear trend of maximum acceleration in certain sub-regions or parameter dimensions of the input space: The first linear function describes the approximate linear negative correlation between the maximum acceleration and the energy-absorbing box within the conventional design range of length (e.g., 200-300mm). The first linear function describes the approximate linear positive correlation between the maximum acceleration and the structural stiffness coefficient over a range of intermediate variations.
[0051] It should be noted that before training the surrogate model of linear combination, this embodiment performs intelligent initialization of the internal parameter set of each basis function in the surrogate model, which facilitates faster optimization convergence and avoids getting trapped in local minima.
[0052] Specifically, the initialization of the internal parameter set of the sigmoid activation function is based on the statistical characteristics of historical datasets to obtain a starting point closest to the optimal solution, including the following steps: Calculate the covariance matrix of the design parameter vectors in the historical dataset and perform principal component analysis to obtain the previous... One principal component and its eigenvalues; number of principal components The determination of this is usually achieved by setting an explanatory variance threshold, such as 85% or 90%, i.e., before selecting... An eigenvector is such that the sum of its corresponding eigenvalues accounts for a proportion of the total variance that reaches a certain threshold.
[0053] Obtain the corresponding random coefficients based on the eigenvalues of each principal component, and then use each random coefficient to... The initial values of the weight vector for each sigmoid activation function are obtained by linearly combining the principal components, as shown in the following formula: , in, Indicates the first The weight vector of each sigmoid activation function. Indicates the first The nth sigmoid activation function generates the first Principal Components eigenvalues The corresponding random coefficients are obtained by randomly sampling from a normal distribution with a mean of zero and a variance equal to the reciprocal of the square root of the eigenvalues of the principal component. .
[0054] In other words, if there are If there are 10 sigmoid activation functions, then a total of 100 sigmoid activation functions need to be generated. Each sigmoid activation function has a set of random coefficients, and its weight vector is determined by its own... Each random coefficient and its corresponding It is composed of linear combinations of principal components.
[0055] Furthermore, based on the weight vector of each sigmoid activation function, the mean vector of the design parameter vector, and the quantile of the physical response value, the initial value of the local bias term for each sigmoid activation function is obtained, as shown in the following formula: , in, Indicates the first Local bias terms of a sigmoid activation function The mean vector of the design parameter vector. This indicates the transpose operation. The quantile represents the physical response value. It is recommended that the value of this quantile be 0.5 (median) or other values that represent the typical response level, such as the 0.25 or 0.75 quantile. This represents the natural logarithm function.
[0056] It should be noted that the principal component-based weight vector initialization ensures that the surrogate model starts from the most prevalent mutation patterns in the learning data; combined with the bias initialization of the quantiles of the physical response values, the activation states of the basis functions are matched with the actual distribution of the data. The combination of these two methods provides a physically meaningful, near-optimal starting point for the optimization process, significantly accelerating subsequent training convergence (reducing the number of iterations) and greatly reducing the risk of the optimization process getting trapped in unfavorable local minima.
[0057] The initialization of the intrinsic parameter set of the Gaussian radial basis functions is determined based on clustering results and includes the following steps: The number of Gaussian radial basis functions is used as the number of clusters, and a clustering algorithm is used to cluster the design parameter vectors in the training dataset; The vector of each cluster center point is used as the initial value of the center point of each Gaussian radial basis function to ensure that the basis functions uniformly cover the entire design space. The initial value of the bandwidth of each Gaussian radial basis function is obtained based on the average distance from the design parameter vector within each cluster to its center point and the scaling factor, as shown in the following formula: , in, and They represent the first The bandwidth and center point of each Gaussian radial basis function; This represents the number of Gaussian radial basis functions. Indicates the center point The first in the cluster A vector of design parameters, Indicates the scaling factor. It can be set to the same value for all Gaussian functions (e.g., 1.2), or it can be individually set according to the statistical characteristics of each cluster, such as density and number of samples.
[0058] It should be noted that by determining the center through clustering and the bandwidth based on intra-cluster statistics, this initialization method fully utilizes the spatial distribution information of the input data. This ensures that the Gaussian radial basis functions are adaptively "deployed" in key regions of the design space, and that the local influence range of each function is adapted to the data density of its region. This provides a crucial prior layout for the Gaussian radial basis functions to accurately capture drastic nonlinear mutations occurring under specific parameter combinations in collision simulations, avoiding problems such as basis function overlap, blanking, or scaling misalignment that may occur with random initialization. This significantly improves the final approximation accuracy and training efficiency of the model.
[0059] For the coefficient vector of a linear function The initialization also utilizes principal component analysis results to ensure that the linear basis functions learn from the direction that best explains the variation in the data. Specifically, the principal components are used as the initial values of the coefficient vectors in each linear function, in descending order of their eigenvalues. If the number of linear functions is less than or equal to the number of principal components, each linear function is assigned an independent principal component; otherwise, the coefficient vectors of the excess linear functions are used cyclically.
[0060] Initializing the linear function with the principal component direction of the data ensures that the linear part can most effectively reconstruct the global linear trend of the data from the beginning of training, providing a stable and accurate linear basis for the entire hybrid model. This significantly reduces the adjustment range and uncertainty of the linear part in the subsequent optimization process, thereby accelerating the convergence of the overall model and improving the stability of the training process.
[0061] This step initializes the internal parameter set of the basis function in multiple ways, forming a multi-level, complementary parameter initialization system that provides a comprehensive and high-quality starting point for the efficient training of the surrogate model.
[0062] S3. Using historical datasets, a two-stage optimization strategy is employed to train the surrogate model, resulting in a well-trained surrogate model used to predict the physical response values corresponding to the new design parameter vectors in CAE simulation tasks.
[0063] It should be noted that the loss function of the surrogate model is the mean squared error between the predicted physical response value and the actual physical response value output by the surrogate model. The goal of training is to find the parameter set that minimizes the loss function. , Represents the first parameter in the parameter set. One parameter.
[0064] The two-stage optimization strategy comprises a local preliminary optimization stage and a global precise optimization stage performed sequentially. The local preliminary optimization stage, the first stage, uses an adaptive learning rate method based on gradient sign to update model parameters. The direction of parameter updates is determined by the sign of the gradient of the loss function, and the update amount (learning rate) is dynamically adjusted based on the consistency of the gradient sign in consecutive iterations. The global precise optimization stage, the second stage, is based on the Gauss-Newton method framework. It obtains the parameter update amount by solving a system of linear equations constructed from the Jacobian matrix and uses an adaptive damping factor to balance convergence speed and stability.
[0065] (1) In each iteration of the first stage, the learning rate is dynamically adjusted according to the adjacent gradient signs of each parameter, so as to obtain the first update amount of each parameter and update it; when the error reduction rate of multiple consecutive iterations is lower than the preset threshold, the first stage ends and the second stage begins.
[0066] Specifically, in the first phase of the In each iteration, the predicted physical response of the current proxy model to all training samples is calculated, and then the gradient of the loss function with respect to each parameter is calculated, and the gradient sign of each parameter is recorded.
[0067] Each parameter is adjusted independently: if the current gradient sign is the same as the previous one, it indicates the optimization direction is consistent, so the learning rate is increased according to the learning rate growth factor; if the sign is opposite, the learning rate is decreased according to the learning rate decay factor; if the gradient is zero, the learning rate remains unchanged. The corresponding learning rate is dynamically adjusted using the following formula: , in, and They represent the first Second and third In the nth iteration Parameters The learning rate; and These represent the preset maximum and minimum learning rates, respectively. and These represent the preset learning rate growth factor and decay factor, respectively. and They represent the first Second and third Loss function in the next iteration and For parameters The gradient; and These represent the minimum value function and the maximum value function, respectively.
[0068] Preferably, the initial learning rate is set to 0.01, the maximum learning rate is set to 0.1, the minimum learning rate is set to 1e-6, the learning rate growth factor is set to 1.2, and the learning rate decay factor is set to 0.5.
[0069] For any parameter The first update value is obtained using the following formula: , in, The sign function is used to output the sign of the gradient (+1, 0, or -1). This means that parameter updates are based solely on the direction of the gradient sign, determined by whether the gradient points to an increase or decrease in loss, and is independent of the magnitude of the gradient. This avoids excessive influence of the gradient magnitude on the optimization process and enhances the stability of the optimization.
[0070] Finally, based on the first update, the updated parameters are: Entering the The next iteration.
[0071] The first stage iteration continues, and the change in the loss function value is monitored. When the rate of decrease of the loss function value is lower than a preset first loss threshold (e.g., 0.1%) after several consecutive iterations (e.g., 10 times), it is considered that the first stage has converged to a relatively stable region, and then it automatically switches to the second stage.
[0072] Preferably, an alternating optimization strategy is adopted, that is, in each iteration: First, fix the internal parameter set of all basis functions. The optimal global bias term is analytically solved using the linear least squares method. and linear weighting coefficients ; Then fix the newly updated global bias. and linear weighting coefficients Using the gradient-sign-based adaptive learning rate method described above, the internal parameters of all basis functions are updated. .
[0073] This alternating strategy makes computation more efficient. Whether it is joint update or alternating update, the core lies in the fact that the first stage completes the preliminary and stable optimization of all parameters.
[0074] (2) In each iteration of the second stage, based on the parameters and surrogate model optimized in the first stage, a linear equation of the Jacobian matrix with adaptive damping factor is constructed and solved to obtain the second update amount of each parameter and update it; when the iteration termination condition is met, the final optimized parameters are obtained.
[0075] This stage employs a damped least squares method based on the Gauss-Newton framework. It obtains parameter updates by constructing and solving a system of linear equations, and introduces an adaptive damping factor to intelligently balance convergence speed and stability.
[0076] Specifically, in the second phase, the In this iteration, the calculation is first performed based on the current parameter set. The Jacobian matrix and error vector.
[0077] Among them, the Jacobian matrix It is The matrix, Indicates the number of training samples. This represents the total number of parameters in the parameter set, and each element in the matrix represents the total number of parameters. , indicating that the model is for the th training samples The output physical response prediction value for the first The partial derivatives of each parameter, i.e., this matrix encapsulates the model's first-order sensitivity information to all parameters.
[0078] Error vector It is The dimension vector is the difference between the actual physical response value and the predicted physical response value of each training sample, i.e., the prediction residual of each training sample.
[0079] Furthermore, a linear equation is constructed to solve for the second update quantity of each parameter, as shown below: , in, Indicates the first The adaptive damping factor in the next iteration is initially set to a small number, such as 0.001. Represents the identity matrix. Indicates the first The second update value of the parameter to be solved in the next iteration is a The vector gives all The adjustment direction and adjustment step size of each parameter.
[0080] It should be noted that the adaptive damping factor is calculated by updating the parameters according to the second update amount obtained in the current iteration. If the loss function value of the surrogate model decreases compared to before the update, the parameters are updated according to the second update amount in the current iteration, and the damping factor is reduced by a coefficient less than 1 (e.g., 0.1) before entering the next iteration; otherwise, the damping factor is increased by a coefficient greater than 1 (e.g., 10), and the second update amount of each parameter in the current iteration is recalculated until the loss function value decreases compared to before the update.
[0081] When the change in the loss function value in the second stage is less than the preset second loss threshold (e.g., 1e-6), or when the maximum number of iterations (e.g., 200) is reached, the second stage of optimization is completed, and each parameter of the final optimization is obtained.
[0082] It should be noted that the second-stage optimization fully utilizes the excellent starting point provided by the first stage. This is achieved by solving for the fused first-order gradient (…). ) and second-order curvature approximation ( The algorithm, which is a linear system, exhibits near-quadratic convergence speed when approaching the optimal solution. Through an adaptive damping factor adjustment mechanism, the algorithm can dynamically and intelligently switch between "fast convergence" and "robust progress," making it suitable for handling highly nonlinear and nonconvex optimization problems in CAE surrogate models. This ensures that while pursuing high accuracy, the optimization process does not diverge or stagnate. This embodiment achieves efficient, stable, and high-accuracy automated training of complex CAE surrogate models through the close integration and synergy of two-stage strategies.
[0083] After training, the resulting surrogate model is a function with a clearly defined mathematical expression. By substituting any new design parameter vector, the predicted physical response can be calculated within milliseconds, significantly improving the efficiency of tasks requiring massive simulation calls, such as simulation-based design optimization and parameter sensitivity analysis.
[0084] Preferably, to further improve the prediction accuracy, robustness, and generalization ability of the model, this embodiment may also employ a hybrid ensemble learning strategy to train the surrogate model. This hybrid ensemble learning strategy includes a aggregation mechanism (Bagging) to reduce variance and a boosting mechanism to correct bias.
[0085] Specifically, the training process of the aggregation mechanism includes: From the historical dataset constructed in step S1, multiple (e.g., 50) training subsets of the same size are generated by sampling with replacement; Based on each training subset, construct the proxy model and initialize the internal parameter set according to step S2, and train the proxy model according to the method of step S3 to obtain multiple proxy models with different structures and parameters, which serve as multiple base learners. By averaging multiple base learners, an initial ensemble model is obtained. ; The new design parameter vector is fed into the initial ensemble model to obtain the initial ensemble output, which is the average of the predictions of each base learner.
[0086] Furthermore, the training process for the enhancement mechanism includes: Calculate the prediction residuals of the current ensemble model (which is the initial ensemble model in the first iteration) on the historical dataset; The prediction residual is used as a new learning objective, and an aggregation mechanism is used to train a residual prediction model. This residual prediction model is obtained by averaging and aggregating multiple (e.g., 30, to control model complexity and computational cost) base residual prediction models. By introducing a shrinkage coefficient, the residual prediction model is superimposed onto the initial ensemble model to form an enhanced model, which is then used as the ensemble model for the next iteration. Repeat the above steps until the maximum number of iterations is reached, or terminate early when the validation set error no longer decreases significantly. The final surrogate model used for prediction is the augmented model after the last iteration, as shown in the following formula: , in, This represents the surrogate model ultimately used for prediction. Indicates the shrinkage coefficient. This is used to control the intensity of each round of enhancement and prevent overfitting; Indicates the total number of iterations. Indicates the first The residual prediction model is obtained through rounds of iteration.
[0087] Compared with existing technologies, the surrogate model construction method for high-dimensional CAE simulation provided in this embodiment constructs a surrogate model through a linear combination of multiple types of basis functions. This enables the surrogate model to possess the inherent ability to uniformly describe linear and nonlinear, global and local features, overcoming the limitation of the expressive power of traditional single-type basis function models. Data-driven initialization places parameters at a high-quality starting point, avoiding optimization from getting trapped in unfavorable local minima from the source. The two-stage optimization achieves a "stable first, then precise" convergence path, making the surrogate model training converge faster and the convergence process more stable and reliable. This significantly reduces the dependence on hyperparameter tuning and the risk of training failure, automatically constructing a surrogate model with both powerful expressive power and efficient training characteristics for complex physical simulation problems, greatly improving prediction accuracy. By using an adaptive method to determine the ratio and number of basis function types, the "self-organization" of the surrogate model structure is achieved, which not only improves the flexibility and accuracy of modeling but also achieves the optimal balance between model complexity and generalization ability through performance comparison on the validation set. The first stage of the two-stage optimization strategy adopts an adaptive learning rate method that relies solely on the gradient sign, which is insensitive to noise and can stably and quickly enter the optimal region. The second stage adopts a higher-order optimization method with an adaptive damping factor, which dynamically adjusts the search strategy according to the error changes and converges quickly when approaching the optimal point. This two-stage collaborative approach enables the entire training process to cope with the non-convexity challenge of the initial stage and complete the final fine-tuning.
[0088] Those skilled in the art will understand that all or part of the processes of the methods described in the above embodiments can be implemented by a computer program instructing related hardware, and the program can be stored in a computer-readable storage medium. The computer-readable storage medium may be a disk, optical disk, read-only memory, or random access memory, etc.
[0089] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for constructing a surrogate model for high-dimensional CAE simulation, characterized in that, Includes the following steps: A historical dataset is constructed based on the design parameter vector in the CAE simulation task and the physical response values obtained through CAE simulation calculation. A proxy model is constructed by linearly combining multiple basis functions of different types. Based on the statistical characteristics of the historical dataset, the internal parameter set of each basis function in the proxy model is initialized; Using the historical dataset, a two-stage optimization strategy is employed to train the surrogate model, resulting in a trained surrogate model used to predict the physical response value corresponding to the new design parameter vector of the CAE simulation task.
2. The surrogate model construction method for high-dimensional CAE simulation according to claim 1, characterized in that, The basis functions include: linear functions, sigmoid activation functions, and Gaussian radial basis functions; the surrogate model is expressed by the following formula: ， in, The proxy model represents the design parameter vector. Predicted physical response value, Indicates the global bias term. Indicates the first basis functions The linear weighting coefficients, Indicates the first basis functions The internal parameter set, This indicates the number of basis functions.
3. The surrogate model construction method for high-dimensional CAE simulation according to claim 2, characterized in that, The type and number of the basis functions are determined adaptively, including: By identifying the relationship patterns between design parameter vectors and physical response values in the historical dataset, the basis function type is adaptively selected. A validation set is partitioned from the historical dataset; based on the initial number of each type of basis function, the final number of each type of basis function is adaptively determined by incrementally constructing and comparing the performance of the proxy model before and after the increment on the validation set.
4. The method for constructing a surrogate model for high-dimensional CAE simulation according to claim 2, characterized in that, The internal parameter set of the sigmoid activation function includes: a weight vector and local bias terms; initializing the internal parameter set of the sigmoid activation function in the surrogate model includes: Calculate the covariance matrix of the design parameter vectors in the historical dataset and perform principal component analysis to obtain the previous... One principal component and its eigenvalues; Based on the eigenvalues of each principal component, corresponding random coefficients are obtained. These random coefficients are then used to... The principal components are linearly combined to obtain the initial values of the weight vector for each sigmoid activation function; The initial values of the local bias terms for each sigmoid activation function are obtained based on the weight vector, the mean vector of the design parameter vector, and the quantile of the physical response value.
5. The surrogate model construction method for high-dimensional CAE simulation according to claim 2, characterized in that, The intrinsic parameter set of the Gaussian radial basis function includes: center point and bandwidth; initialization of the intrinsic parameter set of the Gaussian radial basis function in the surrogate model includes: The number of Gaussian radial basis functions is used as the number of clusters, and a clustering algorithm is used to cluster the design parameter vectors in the training dataset; The vector of each cluster center point is used as the initial value of the center point of each Gaussian radial basis function; the initial value of the bandwidth of each Gaussian radial basis function is obtained based on the average distance from the design parameter vector within each cluster to its center point and the scaling factor.
6. The surrogate model construction method for high-dimensional CAE simulation according to claim 1, characterized in that, The two-stage optimization strategy includes: In each iteration of the first stage, the learning rate is dynamically adjusted according to the adjacent gradient signs of each parameter, thereby obtaining the first update amount of each parameter and updating it; when the error reduction rate of multiple consecutive iterations is lower than a preset threshold, the first stage ends and the second stage begins. In each iteration of the second stage, based on the parameters and surrogate model optimized in the first stage, a linear equation for the Jacobian matrix with an adaptive damping factor is constructed and solved to obtain the second update amount for each parameter and update it; when the iteration termination condition is met, the final optimized parameters are obtained.
7. The surrogate model construction method for high-dimensional CAE simulation according to claim 6, characterized in that, The learning rate is dynamically adjusted based on the adjacent gradient signs of each parameter using the following formula: ， in, and They represent the first Second and third In the nth iteration Parameters The learning rate; and These represent the preset maximum and minimum learning rates, respectively. and These represent the preset learning rate growth factor and decay factor, respectively. and They represent the first Second and third Loss function in the next iteration and For parameters The gradient; and These represent the minimum value function and the maximum value function, respectively.
8. The method for constructing a surrogate model for high-dimensional CAE simulation according to claim 6, characterized in that, The adaptive damping factor is calculated by updating the loss function value of the surrogate model after the parameters are updated according to the second update amount obtained in the current iteration. If the loss function value decreases compared to before the update, the damping factor is reduced by a coefficient less than 1 after updating the parameters according to the second update amount in the current iteration, and then the next iteration is entered. Otherwise, increase the damping factor by a coefficient greater than 1 and recalculate the second update amount in the current iteration until the loss function value is reduced compared to before the update.
9. The method for constructing a surrogate model for high-dimensional CAE simulation according to claim 4, characterized in that, The random coefficients obtained based on the eigenvalues of each principal component are obtained by random sampling from a normal distribution with a mean of zero and a variance equal to the reciprocal of the square root of the eigenvalues of the principal components.
10. The method for constructing a surrogate model for high-dimensional CAE simulation according to claim 4, characterized in that, The initial values of the local bias terms for each sigmoid activation function are obtained using the following formula: ， in, Indicates the first Local bias terms of a sigmoid activation function Indicates the first The weight vector of each sigmoid activation function. The mean vector of the design parameter vector. Indicates the quantiles of the physical response value. This indicates the transpose operation.