Adaptive hierarchical Kriging model construction method based on sequential alternate sampling strategy

Abstract

The invention discloses a self-adaptive layered Kriging model construction method based on a sequential alternate sampling strategy, and belongs to the technical field of multi-fidelity agent modeling. The method comprises the following steps: firstly, generating an initial input sample for each model with different fidelity, calculating the response of the fidelity model, generating a plurality of input samples, establishing an initial Kriging model and an initial layered Kriging model, and establishing a low-fidelity training sample set and a high-fidelity training sample set; and setting an initial iteration frequency, and calculating an initial cost budget. Secondly, judging whether a stopping condition of total cost budget is met or not, if yes, outputting a final layered Kriging model in the last step, and if not, iterating again; according to the method, through a self-adaptive sampling point distribution mechanism, the sample sizes required by the high-fidelity model and the low-fidelity model can be dynamically and intelligently determined, and the modeling efficiency is improved; the method can automatically recognize a function feature region with a significant maximum value and a significant minimum value, and reduces the overall data collection cost to the maximum extent while guaranteeing the final meta-model precision.

Description

An Adaptive Hierarchical Kriging Model Construction Method Based on Sequential Alternating Sampling Strategy Technical Field

[0001] This invention belongs to the field of multi-fidelity proxy modeling technology, and relates to an adaptive sample allocation method for simulation and analysis of complex structures, and more particularly to an adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy. Background Technology

[0002] Multi-fidelity data is widely used in various fields such as engineering optimization, physical simulation, and financial modeling. High-fidelity data (such as high-precision experimental measurements or high-resolution numerical simulation results) offers high accuracy but often comes with extremely high time or economic costs; conversely, low-fidelity data (such as simplified model simulations or historical experience data) is inexpensive, but its accuracy and reliability are relatively limited. To achieve a balance between accuracy and cost, multi-fidelity proxy modeling technology has emerged. Its core idea is to construct a meta-model that combines high accuracy and low cost by fusing data of different fidelities. Current research on multi-fidelity proxy modeling technology focuses on the fusion strategies of multi-fidelity models, while a systematic method for sampling point allocation is still immature. Current mainstream sampling point allocation methods mainly rely on user experience or multiple trial-and-error adjustments, i.e., manually presetting the number of sampling points for each fidelity model and then repeatedly correcting it based on the modeling results.

[0003] To address the challenges of efficient sampling and accuracy improvement in complex models, scholars both domestically and internationally have conducted extensive research and proposed several adaptive sequential experimental design methods. For example, Chinese invention patent (application number 202211510849.4) provides an adaptive sequential experimental design method based on a hybrid addition criterion. This method performs initial experimental design on the parameters influencing the mission performance indicators of aerospace equipment systems, fits the data using a least squares support vector machine model, and performs sequential sampling based on the LOLA-DIST addition criterion to optimize the design space distribution, thereby improving prediction accuracy and efficiency. Another Chinese invention patent (application number 202411776747.6) provides a sequential experimental design method for spacecraft based on the expected probability box boundary improvement criterion. This method uses Gaussian process regression to construct a surrogate model and handles uncertainties in spacecraft experiments through the probability box boundary improvement criterion. It prioritizes sampling in the extreme value region of the objective function to quickly converge to the optimal solution, verifying the effectiveness of this method in aerodynamic and thermal environment prediction. While existing technologies enhance the local development capabilities and global exploratory power of surrogate models through adaptive sampling strategies, most are limited to single-fidelity models. They fail to fully utilize multi-fidelity data to balance the high cost of high-fidelity models with the efficiency advantages of low-fidelity models. Furthermore, they lack a dynamic alternating sampling mechanism to adaptively switch the sampling focus of different fidelity models, making it difficult to achieve the optimal trade-off between cost and accuracy in complex system optimization. Therefore, there is an urgent need for an adaptive multi-fidelity model construction method that can integrate a sequential alternating sampling strategy to address the shortcomings of existing technologies in multi-source data fusion and resource optimization. Summary of the Invention

[0004] To address the problems existing in the prior art, this invention provides an adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy. This method enables the dynamic and efficient allocation of sampling budgets to each fidelity model without relying on prior experience, thereby minimizing the total sampling cost while ensuring the accuracy of the multi-fidelity surrogate model.

[0005] To solve the above technical problems, the technical solution adopted by the present invention is: an adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy, referred to as the SASS method. The adaptive hierarchical Kriging model construction method includes the following steps: Step S1: In the existing modeling and simulation environment, by setting the design variables and operating parameters of the target system, the corresponding high-fidelity model, low-fidelity model and sample space of input variables are obtained.

[0006] Step S2: Generate initial input samples for each model with different fidelity levels, and calculate the response of the corresponding fidelity model at different fidelity levels. Use the Latin hypercube sampling method to generate low-fidelity initial input samples x. l(Generated from a low-fidelity model), high-fidelity initial input sample x h (Generated from a high-fidelity model), high-fidelity verification input sample x t (Generated from a high-fidelity model).

[0007] Step S3: Based on the low-fidelity initial input sample x l and the corresponding output response f l Establish the initial Kriging model. Based on the high-fidelity initial input sample x h The corresponding output response f h An initial hierarchical Kriging model is established using the low-fidelity Kriging model. A low-fidelity training sample set D is then created. l =[x l ,f l High-fidelity training sample set D h =[x h ,f h ]. Based on high-fidelity verification of input sample x t The corresponding output response f t Establish a high-fidelity verification sample set D t =[x t ,f t Set the initial iteration count to id=1. Calculate the initial cost budget CT0=N. h0 +N l0 ·C, where N h0 N represents the number of initial high-fidelity input samples. l0 This represents the number of initial input samples for low-fidelity simulation, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

[0008] Step S4: Check if the current cost budget meets the stopping condition for the total cost budget. Set the stopping condition to the total cost budget. Once the calculated budget is exhausted, proceed to step S8 and output the final hierarchical Kriging model; otherwise, proceed to step S5. The total cost budget CT is: (1) In the formula, N l N represents the number of low-fidelity samples. h denoted by , where represents the number of high-fidelity samples, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

[0009] Step S5: If the iteration number id is odd, use the MEFmin acquisition function to obtain updated samples, and let id = id + 1. Otherwise, proceed to step S6. This mainly involves comparing the MEF values ​​corresponding to different fidelities. minThe maximum value determines the current updated sample and the corresponding fidelity source. The steps to obtain the updated sample are as follows: Step S5-1: Consider the uncertainty of the low-fidelity model near the minimum value. In the low-fidelity model, the response of the input updated sample x... It follows a Gaussian distribution. After standardization, its expression is: (2) In the formula, This represents the predicted response result for the i-th sample. This represents the predicted mean of the i-th sample. This represents the prediction standard deviation of the i-th sample. Represents a variable that follows a standard normal distribution. This represents a standard normal distribution.

[0010] Step S5-2: Consider the predicted value of the response The improvement at the minimum value is defined as: (3) In the formula, I represents the improvement effect. This represents the minimum value of the current high-fidelity sample. This represents the predicted response result for the i-th sample. This represents the predicted mean of the i-th sample. Let z represent the predicted standard deviation of the i-th sample, and z represent the variable that follows a standard normal distribution.

[0011] Step S5-3: Let Then we further obtain: (4) Step S5-4: Calculate the expectation of I, and its improved expression is: In formula (5), This represents the expected improvement effect of I. Let z represent the standard normal probability density function. , Let z0 be the standard normal probability density function.

[0012] Step S5-5: Further extend to different fidelity cases, including low fidelity and high fidelity, to obtain: (6) In the formula, tt represents the high-fidelity source, h represents the high-fidelity source, and l represents the low-fidelity source. This represents the prediction standard deviation of the high-fidelity model. This represents the minimum value of the high-fidelity model. This represents the predicted mean of the high-fidelity model. This represents the prediction standard deviation of the low-fidelity model. This represents the minimum value of the low-fidelity model. This represents the predicted mean of the low-fidelity model.

[0013] Step S5-6: Define correlation functions for different fidelity levels, mainly considering the predictive correlation functions of low-fidelity and high-fidelity models. The specific expressions for these functions are as follows: In equation (7), This represents the correlation function.

[0014] Step S5-7: Considering the differences in calculation costs for samples with different fidelity, the specific expression of the cost function is defined as follows: In formula (8), Represents the cost function, C h C represents the computational cost of high-fidelity samples. l This indicates the computational cost of low-fidelity samples.

[0015] Step S5-8: To avoid the updated samples being too close to the high-fidelity samples, resulting in overly dense sample clusters and thus reducing the prediction performance of the high-fidelity model, a sample density function is defined. for: (9) In the formula, N h N represents the number of high-fidelity samples. total S represents the total number of high-fidelity samples and low-fidelity samples. h S represents a high-fidelity sample. total This represents the set of high-fidelity samples and low-fidelity samples. Represents x and the i-th S h Gaussian correlation function, Let x represent the i-th S. total The Gaussian correlation function is used. The reason for this setting is that when the updated sample is a low-fidelity sample, if the updated sample is the same as the high-fidelity sample, it is meaningless for improving the accuracy of the high-fidelity model. Therefore, it is necessary to ensure that the updated sample is far away from both high-fidelity and low-fidelity samples.

[0016] Step S5-9: Considering the uncertainties of the low-fidelity and high-fidelity models near the minimum, the acquisition function is established as follows: (10) Step S5-10: Updating the samples and the corresponding fidelity can be achieved by minimizing MEF. min To determine, as defined by the following formula: In equation (11), x new Indicates updating the sample, tt new This indicates the fidelity source (low fidelity or high fidelity) of the updated sample, with different fidelity values ​​corresponding to different MEF values. min Maximum value search is achieved using the particle swarm optimization algorithm. The MEF values ​​corresponding to different fidelities are compared. minThe maximum value determines the current update sample and the corresponding fidelity source. That is, two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively, and then the sample with the larger acquisition function value is selected as the update sample in step S5.

[0017] Step S5-11: Based on fidelity tt new Select the appropriate model, if tt new =h, then choose the high-fidelity model; if tt new If the value is 1, then the low-fidelity model is selected. Update sample x. new The input is fed into the corresponding model to obtain the output response. If tt new =h, then update the sample to [x new ,f hnew If tt new =h, then update the sample to [x new ,f lnew ], where f hnew x represents new The output response, f, in the high-fidelity model lnew x represents new Output response in a low-fidelity model.

[0018] Step S6: If the iteration number id is even, use the MEFmax acquisition function to obtain updated samples, and let id = id + 1. The steps to obtain updated samples are as follows: This step mainly involves comparing the MEF values ​​corresponding to different fidelities. max The maximum value determines the current updated sample and the corresponding fidelity source. Specifically: Step S6-1: Consider the improvement effect of the predicted response at the maximum value, and define: In equation (12), I2 represents the improvement effect. This represents the maximum value of the current high-fidelity sample.

[0019] Step S6-2: Let Then we further obtain: (13) Step S6-3: Calculate the expectation of I2, and its improved expression is: In equation (14), This represents the expected improvement effect of I2. , Indicate z 02 The standard normal probability density function.

[0020] Step S6-4: Further extend to different fidelity cases, including low fidelity and high fidelity, to obtain: In equation (15), This represents the maximum value of the high-fidelity model. This represents the maximum value of the low-fidelity model.

[0021] Step S6-5: Considering the uncertainties of the low-fidelity and high-fidelity models near their maximum values, the acquisition function established by combining the correlation function, cost function, and sample density function is as follows: (16) Step S6-6: Updating the samples and the corresponding fidelity can be achieved by maximizing MEF. max To determine, as defined by the following formula: In equation (17), x new Indicates updating the sample, tt new This indicates the fidelity source (low fidelity or high fidelity) of the updated sample, with different fidelity values ​​corresponding to different MEF values. max Maximum value search is achieved using the particle swarm optimization algorithm. The MEF values ​​corresponding to different fidelities are compared. max The maximum value determines the current update sample and the corresponding fidelity source. That is, two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively, and then the sample with the larger acquisition function value is selected as the update sample in step S6.

[0022] Steps S6-7: Based on fidelity tt new Select the appropriate model, if tt new =h, then choose the high-fidelity model; if tt new If the value is 1, then the low-fidelity model is selected. Update sample x. new The input is fed into the corresponding model to obtain the output response. If tt new =h, then update the sample to [x new ,f hnew If tt new =h, then update the sample to [x new ,f lnew ], where f hnew x represents new The output response, f, in the high-fidelity model lnew x represents new Output response in a low-fidelity model.

[0023] Step S7: Update low-fidelity sample D l =D l ∪[x new ,f lnew Or high-fidelity sample D h =D h ∪[x new ,f hnew Return to step S4.

[0024] Step S8: Using D l and D h Output the final hierarchical Kriging model and validate it against high-fidelity sample D.t Accuracy prediction is performed to obtain error assessment results for the maximum relative error (MAE) and root mean square error (RMSE).

[0025] The beneficial effects of this invention are as follows: First, this invention effectively overcomes the problems of low efficiency and reliance on prior experience in sampling strategies in the prior art. Its beneficial effect lies in the fact that through an adaptive sampling point allocation mechanism, it can dynamically and intelligently determine the sample size required for high-fidelity and low-fidelity models, avoiding tedious trial-and-error processes and significantly improving modeling efficiency. Second, this invention can automatically identify function feature regions with significant maxima and minima, and prioritize the collection of low-cost data in these regions, thereby maximizing the reduction of overall data collection costs while ensuring the accuracy of the final meta-model. Finally, this invention provides a robust and automated construction method for achieving high-precision, low-cost hierarchical Kriging proxy modeling. Attached Figure Description

[0026] Figure 1 is a flowchart of an embodiment of the present invention; Figure 2 is a comparison diagram of error results in Embodiment 1 of the present invention; Figure 2(a) is a comparison diagram of MAE error results; Figure 2(b) is a comparison diagram of RMSE error results; Figure 3 is a diagram of different fidelity models in Embodiment 2 of the present invention; Figure 3(a) is a high-fidelity model; Figure 3(b) is a low-fidelity model; Figure 4 is a comparison diagram of displacement and strength prediction results in Embodiment 2 of the present invention; Figure 4(a) is a comparison diagram of displacement prediction results; Figure 4(b) is a comparison diagram of stress prediction results. Detailed Implementation

[0027] To make the technical solutions and advantages of the present invention clearer, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are used to illustrate the present invention, but should not be used to limit the scope of the present invention.

[0028] Example 1: As shown in Figure 1, this example provides an adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy, including the following steps: Step S1: In the existing modeling and simulation environment, by setting the design variables and operating parameters of the target system, the corresponding high-fidelity model, low-fidelity model, and sample space of the input variables are obtained. The expressions for the high-fidelity model and the low-fidelity model are as follows: (18) Step S2: Generate initial input samples for each model with different fidelity, and calculate the corresponding model response at different fidelity levels. Six low-fidelity initial input samples x are generated using the Latin hypercube sampling method. l (Generated from a low-fidelity model), 3 high-fidelity initial input samples x h (Generated from a high-fidelity model), 100 high-fidelity validation input samples xt (Generated from a high-fidelity model).

[0029] Step S3: Based on the low-fidelity initial input sample x l and the corresponding output response f l Establish the initial Kriging model. Based on the high-fidelity initial input sample x h The corresponding output response f h An initial hierarchical Kriging model is established using the low-fidelity Kriging model. A low-fidelity training sample set D is then created. l =[x l ,f l High-fidelity training sample set D h =[x h ,f h ]. Based on high-fidelity verification of input sample x t The corresponding output response f t Establish a high-fidelity verification sample set D t =[x t ,f t Set the initial iteration count to id=1. Calculate the initial cost budget CT0=N. h0 +N l0 ·C, where N h0 N represents the number of initial high-fidelity input samples. l0 This represents the number of initial input samples for low-fidelity simulation, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

[0030] Step S4: Check if the current cost budget meets the stopping condition for the total cost budget. Set the stopping condition to the total cost budget. Once the calculated budget is exhausted, proceed to step S8 and output the final hierarchical Kriging model; otherwise, proceed to step S5. The total cost budget CT is... (1) In the formula, N l N represents the number of low-fidelity samples. h denoted by , where represents the number of high-fidelity samples, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

[0031] Step S5: If the iteration number id is odd, use the MEFmin acquisition function to obtain the updated sample, and let id = id + 1. Otherwise, proceed to step 6. The steps to obtain the updated sample are as follows: Step S5-1: Consider the uncertainty of the low-fidelity model near the minimum. In the low-fidelity model, the response of the input updated sample x... It follows a Gaussian distribution. After standardization, its expression is: (2) In the formula, This represents the predicted response result for the i-th sample. This represents the predicted mean of the i-th sample. This represents the prediction standard deviation of the i-th sample. Represents a variable that follows a standard normal distribution. This represents a standard normal distribution.

[0032] Step S5-2: Consider the predicted value of the response The improvement at the minimum value is defined as: (3) In the formula, I represents the improvement effect. This represents the minimum value of the current high-fidelity sample. This represents the predicted response result for the i-th sample. This represents the predicted mean of the i-th sample. Let z represent the predicted standard deviation of the i-th sample, and z represent the variable that follows a standard normal distribution.

[0033] Step S5-3: Let Then we further obtain: (4) Step S5-4: Calculate the expectation of I, and its improved expression is: In formula (5), This represents the expected improvement effect of I. Let z represent the standard normal probability density function. , Let z0 be the standard normal probability density function.

[0034] Step S5-5: Further extending to different fidelity cases (i.e., low fidelity and high fidelity), we obtain: (6) In the formula, tt represents the high-fidelity source, h represents the high-fidelity source, and l represents the low-fidelity source. This represents the prediction standard deviation of the high-fidelity model. This represents the minimum value of the high-fidelity model. This represents the predicted mean of the high-fidelity model. This represents the prediction standard deviation of the low-fidelity model. This represents the minimum value of the low-fidelity model. This represents the predicted mean of the low-fidelity model.

[0035] Step S5-6: Define correlation functions for different fidelity levels, mainly considering the predictive correlation functions of low-fidelity and high-fidelity models. The specific expressions for these functions are as follows: In equation (7), This represents the correlation function.

[0036] Step S5-7: Considering the differences in cost calculation for samples with different fidelity, the specific expression of the cost function is defined as follows: In formula (8), Represents the cost function, C hC represents the computational cost of high-fidelity samples. l This indicates the computational cost of low-fidelity samples.

[0037] Step S5-8: To avoid the updated samples being too close to the high-fidelity samples, resulting in overly dense sample clusters and thus reducing the prediction performance of the high-fidelity model, a sample density function is defined. for (9) In the formula, N h N represents the number of high-fidelity samples. total S represents the total number of high-fidelity samples and low-fidelity samples. h S represents a high-fidelity sample. total This represents the set of high-fidelity samples and low-fidelity samples. Represents x and the i-th S h Gaussian correlation function, Let x represent the i-th S. total The Gaussian correlation function is used. The reason for this setting is that when the updated sample is a low-fidelity sample, if the updated sample is the same as the high-fidelity sample, it is meaningless for improving the accuracy of the high-fidelity model. Therefore, it is necessary to ensure that the updated sample is far away from both high-fidelity and low-fidelity samples.

[0038] Step S5-9: Considering the uncertainties of the low-fidelity and high-fidelity models near the minimum, the established acquisition function is as follows: (10) Step S5-10: Updating the samples and the corresponding fidelity can be achieved by minimizing MEF. min To determine, as defined by the following formula: In equation (11), x new Indicates updating the sample, tt new This indicates the fidelity source (low fidelity or high fidelity) of the updated sample, with different fidelity values ​​corresponding to different MEF values. min Maximum value search is achieved using the particle swarm optimization algorithm. The MEF values ​​corresponding to different fidelities are compared. min The maximum value determines the current update sample and the corresponding fidelity source. That is, two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively, and then the sample with the larger acquisition function value is selected as the update sample in step S5.

[0039] Step S5-11: Based on fidelity tt new Select the appropriate model, if tt new =h, then choose the high-fidelity model; if tt new If the value is 1, then the low-fidelity model is selected. Update sample x. new The input is fed into the corresponding model to obtain the output response. If tt new =h, then update the sample to [x new ,fhnew If tt new =h, then update the sample to [x new ,f lnew ], where f hnew x represents new The output response, f, in the high-fidelity model lnew x represents new Output response in a low-fidelity model.

[0040] Step S6: If the iteration number id is even, use the MEFmax acquisition function to obtain updated samples, and let id = id + 1. The steps to obtain updated samples are as follows: Step S6-1: Consider the improvement effect of the predicted response value at the maximum value, and define: In equation (12), I2 represents the improvement effect. This represents the maximum value of the current high-fidelity sample.

[0041] Step S6-2: Let Then we further obtain: (13) Step S6-3: Calculate the expectation of I2, then its improved expression is as follows: In equation (14), This represents the expected improvement effect of I2. , Indicate z 02 The standard normal probability density function.

[0042] Step S6-4: Further extending to different fidelity cases (i.e., low fidelity and high fidelity), we further obtain: In equation (15), This represents the maximum value of the high-fidelity model. This represents the maximum value of the low-fidelity model.

[0043] Step S6-5: Considering the uncertainties of the low-fidelity and high-fidelity models near their maximum values, the acquisition function established by combining the correlation function, cost function, and sample density function is as follows: (16) Step S6-6: Updating the samples and the corresponding fidelity can be achieved by maximizing MEF. max To determine, as defined by the following formula: In equation (17), x new Indicates updating the sample, tt new This indicates the fidelity source (low fidelity or high fidelity) of the updated sample, with different fidelity values ​​corresponding to different MEF values. max Maximum value search is achieved using the particle swarm optimization algorithm. The MEF values ​​corresponding to different fidelities are compared. maxThe maximum value determines the current update sample and the corresponding fidelity source. That is, two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively, and then the sample with the larger acquisition function value is selected as the update sample in step S6.

[0044] Steps S6-7: Based on fidelity tt new Select the appropriate model, if tt new =h, then choose the high-fidelity model; if tt new If the value is 1, then the low-fidelity model is selected. Update sample x. new The input is fed into the corresponding model to obtain the output response. If tt new =h, then update the sample to [x new ,f hnew If tt new =h, then update the sample to [x new ,f lnew ], where f hnew x represents new The output response, f, in the high-fidelity model lnew x represents new Output response in a low-fidelity model.

[0045] Step S7: Update low-fidelity sample D l =D l ∪[x new ,f lnew Or high-fidelity sample D h =D h ∪[x new ,f hnew Return to step S4.

[0046] Step S8: Using D l and D h Output the final hierarchical Kriging model and validate it against high-fidelity sample D. t Accuracy prediction was performed, and the error evaluation results of maximum relative error (MAE) and root mean square error (RMSE) were obtained. The proposed method was compared with the Variable Fidelity Expectation Improvement Method (VFEI), the Adaptive Multi-Fidelity Expectation Improvement Method (AMEI), and the Two-Function Adaptive Selection Method (ASDM). All methods used the same initial low-fidelity training samples, initial low-fidelity training samples, and high-fidelity validation samples. The resulting beehive diagram of the error results is shown in Figure 2. It can be seen that among all methods, the proposed method yields the smallest MAE and RMSE errors. With a fixed cost, the proposed method can achieve higher prediction accuracy.

[0047] Example 2: As shown in Figure 1, this example provides an adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy, including the following steps: Step S1: Consider the strength and displacement requirements of the aircraft segment under operating conditions. In the existing modeling and simulation environment, by setting the design variables and operating condition parameters of the aircraft segment, the corresponding high-fidelity model, low-fidelity model, and sample space of the input variables are obtained. Among them, the high-fidelity model and the low-fidelity model are shown in Figure 3, where Figure 3(a) is the high-fidelity model and Figure 3(b) is the low-fidelity model.

[0048] Step S2: Generate initial input samples for each model with different fidelity levels, and calculate the corresponding model responses at different fidelity levels. Use the Latin hypercube sampling method to generate 100 low-fidelity initial input samples x. l (Generated from a low-fidelity model), 35 high-fidelity initial input samples x h (Generated from a high-fidelity model), 50 high-fidelity validation input samples x t (Generated from a high-fidelity model).

[0049] Step S3: Based on the low-fidelity initial input sample x l and the corresponding output response f l Establish the initial Kriging model. Based on the high-fidelity initial input sample x h The corresponding output response f h An initial hierarchical Kriging model is established using the low-fidelity Kriging model. A low-fidelity training sample set D is then created. l =[x l ,f l High-fidelity training sample set D h =[x h ,f h ]. Based on high-fidelity verification of input sample x t The corresponding output response f t Establish a high-fidelity verification sample set D t =[x t ,f t Set the initial iteration count to id=1. Calculate the initial cost budget CT0=N. h0 +N l0 ·C, where N h0 N represents the number of initial high-fidelity input samples. l0 This represents the number of initial input samples for low-fidelity simulation, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

[0050] Step S4: Check if the current cost budget meets the stopping condition for the total cost budget. Set the stopping condition to the total cost budget. Once the calculated budget is exhausted, proceed to step S8 and output the final hierarchical Kriging model; otherwise, proceed to step S5. The total cost budget CT is... (1) In the formula, N l N represents the number of low-fidelity samples. h denoted by , where represents the number of high-fidelity samples, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

[0051] Step S5: If the iteration number id is odd, use the MEFmin collection function to obtain the updated sample, and let id = id + 1. Otherwise, proceed to step 6.

[0052] Step S6: If the iteration number id is even, use the MEFmax acquisition function to obtain the updated sample, and let id = id + 1.

[0053] Step S7: Update low-fidelity sample D l =D l ∪[x new ,f lnew Or high-fidelity sample D h =D h ∪[x new ,f hnew Return to step S4.

[0054] Step S8: Using D l and D h Output the final hierarchical Kriging model and validate it against high-fidelity sample D. t Accuracy prediction is performed to obtain error assessment results for the maximum relative error (MAE) and root mean square error (RMSE).

[0055] Verification of the results in this embodiment: The method in this embodiment is compared with the Variable Fidelity Expectation Improvement Method (VFEI), the Adaptive Multi-Fidelity Expectation Improvement Method (AMEI), and the Two-Function Adaptive Selection Method (ASDM). All methods use the same initial low-fidelity training samples, initial low-fidelity training samples, and high-fidelity verification samples. The final displacement and stress error results obtained by the method in this embodiment and the comparison methods are shown in Figure 4 (Figure 4(a) is a comparison of displacement prediction results, and Figure 4(b) is a comparison of stress prediction results), Table 1, and Table 2.

[0056] Table 1: Displacement Prediction Results

[0057] Table 2: Stress Prediction Results

[0058] It can be seen that among all methods, the MAE error and RMSE error obtained by this invention are the smallest. With a fixed cost, the proposed method can achieve higher accuracy in displacement and stress prediction.

[0059] The above embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. An adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy, characterized in that, The adaptive hierarchical Kriging model construction method includes the following steps: Step S1: By setting the design variables and operating parameters of the target system, obtain the corresponding high-fidelity model, low-fidelity model, and sample space of the input variables; Step S2: Generate initial input samples for each model with different fidelity, and calculate the response of the corresponding fidelity model at different fidelity levels; and generate low-fidelity initial input samples x respectively. l High-fidelity initial input sample x h High-fidelity verification of input sample x t Step S3: Based on the low-fidelity initial input sample x l and the corresponding output response f l Establish an initial Kriging model; based on high-fidelity initial input samples x h The corresponding output response f h An initial hierarchical Kriging model is established using a low-fidelity Kriging model; a low-fidelity training sample set D is established. l =[x l ,f l High-fidelity training sample set D h =[x h ,f h Based on high-fidelity verification of input sample x t The corresponding output response f t Establish a high-fidelity verification sample set D t =[x t ,f t Set the initial iteration count to id=1; calculate the initial cost budget CT0=N. h0 +N l0 ·C, where N h0 N represents the number of initial high-fidelity input samples. l0 represents the initial number of low-fidelity input samples, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation; Step S4: Check whether the current cost budget meets the stopping condition of the total cost budget; Set the stopping condition to the total cost budget. When the calculation budget is exhausted, proceed to step S8 and output the final hierarchical Kriging model; otherwise, proceed to step S5; Step S5: If the iteration number id is odd, use the MEFmin acquisition function to obtain updated samples and let id = id + 1; otherwise, proceed to step S6; By comparing the MEF values ​​corresponding to different fidelities... min The maximum value determines the current update sample and the corresponding fidelity source. Two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively. The sample with the larger acquisition function value is selected as the update sample in step S5. Step S6: If the iteration number id is even, the update sample is obtained using the MEFmax acquisition function, and id = id + 1 is set. By comparing the MEF values ​​corresponding to different fidelities... max The maximum value determines the current update sample and the corresponding fidelity source. Two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively. The sample with the larger acquisition function value is selected as the update sample in step S6. Step S7: Update the low-fidelity sample D. l =D l ∪[x new ,f lnew Or high-fidelity sample D h =D h ∪[x new ,f hnew Return to step S4; Step S8: Utilize D l and D h Output the final hierarchical Kriging model and validate it against high-fidelity sample D. t Perform accuracy prediction.

2. The adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy according to claim 1, characterized in that, In step S2, the Latin hypercube sampling method is used to generate low-fidelity initial input samples, high-fidelity initial input samples, and high-fidelity verification input samples, respectively.

3. The adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy according to claim 1, characterized in that, In step S4, the total cost budget CT is: (1); where N l N represents the number of low-fidelity samples. h denoted by , where represents the number of high-fidelity samples, and C represents the cost ratio between low-fidelity model simulation and high-fidelity model simulation.

4. The adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy according to claim 1, characterized in that, In step S5, the steps to obtain the updated sample are as follows: Step S5-1: In the low-fidelity model, input the response of the updated sample x. It follows a Gaussian distribution. After standardization, its expression is: (2); where, This represents the predicted response result for the i-th sample. This represents the predicted mean of the i-th sample. This represents the prediction standard deviation of the i-th sample. Represents a variable that follows a standard normal distribution. Represents a standard normal distribution; Step S5-2: Consider the predicted value of the response. The improvement at the minimum value is defined as: (3); where I represents the improvement effect, This represents the minimum value of the current high-fidelity sample. This represents the predicted response result for the i-th sample. This represents the predicted mean of the i-th sample. Let represent the predicted standard deviation of the i-th sample, and z represent a variable that follows a standard normal distribution; Step S5-3: Let ,get: (4); Step S5-4: Calculate the expectation of I, then its improved expression for expectation is: (5); where, This represents the expected improvement effect of I. Let z represent the standard normal probability density function. ， Let z0 be the standard normal probability density function; Step S5-5: Extend to different fidelity cases, including low fidelity and high fidelity, to obtain: (6); where tt represents the high-fidelity source, h represents the high-fidelity source, and l represents the low-fidelity source. This represents the prediction standard deviation of the high-fidelity model. This represents the minimum value of the high-fidelity model. This represents the predicted mean of the high-fidelity model. This represents the standard deviation of the predictions of the low-fidelity model. This represents the minimum value of the low-fidelity model. This represents the prediction mean of the low-fidelity model; Step S5-6: Define the correlation function for different fidelity levels. Considering the prediction correlation function of the low-fidelity model and the high-fidelity model, the specific expressions are as follows: (7); where, Represents the correlation function; Step S5-7: Considering the differences in calculation costs for samples with different fidelity, the specific expression of the cost function is defined as follows: (8); where, Represents the cost function, C h C represents the computational cost of high-fidelity samples. l Indicates the computational cost of low-fidelity samples; Step S5-8: Define the sample density function for: (9); where N h N represents the number of high-fidelity samples. total S represents the total number of high-fidelity samples and low-fidelity samples. h S represents a high-fidelity sample. total This represents the set of high-fidelity samples and low-fidelity samples; Represents x and the i-th S h Gaussian correlation function, Let x represent the i-th S. total The Gaussian correlation function is used; the reason for this setting is that when the updated sample is a low-fidelity sample, if the updated sample overlaps with the high-fidelity sample, it is meaningless for improving the accuracy of the high-fidelity model. Therefore, it is necessary to ensure that the updated sample is far away from both high-fidelity and low-fidelity samples. Step S5-9: Considering the uncertainty of the low-fidelity model and the high-fidelity model near the minimum value, the acquisition function is established as follows: (10); Steps S5-10: Updating the samples and the corresponding fidelity can be achieved by minimizing the MEF. min To determine, as defined by the following formula: (11); where x new Indicates updating the sample, tt new This represents the fidelity source for updating samples, with different fidelities corresponding to different MEF values. min Maximum value search is achieved using particle swarm optimization algorithm; the MEF values ​​corresponding to different fidelities are compared. min The maximum value determines the current update sample and the corresponding fidelity source. Two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively. The sample with the larger acquisition function value is selected as the update sample in step S5. Step S5-11: Based on the fidelity tt new Select the appropriate model, if tt new =h, then choose the high-fidelity model; if tt new =l, then select the low-fidelity model; update sample x new The input is fed into the corresponding model to obtain the output response; if tt new =h, then update the sample to [x new ,f hnew If tt new =h, then update the sample to [x new ,f lnew ], where f hnew x represents new The output response, f, in the high-fidelity model lnew x represents new Output response in a low-fidelity model.

5. The adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy according to claim 1, characterized in that, In step S6, the steps to obtain the updated sample are as follows: Step S6-1: Considering the improvement effect of the predicted response value at the maximum value, define: (12); where I2 represents the improvement effect, Indicates the maximum value of the current high-fidelity sample; Step S6-2: Let Then we get: (13); Step S6-3: Calculate the expectation of I2, and its improved expression is: (14); where, This represents the expected improvement effect of I2. ， Indicate z 02 The standard normal probability density function; Step S6-4: Extend to different fidelity cases, including low fidelity and high fidelity, to obtain: (15); where, This represents the maximum value of the high-fidelity model. This represents the maximum value of the low-fidelity model; Step S6-5: Considering the uncertainties of the low-fidelity and high-fidelity models near the maximum value, the acquisition function established by combining the correlation function, cost function, and sample density function is as follows: (16); Step S6-6: Updating the samples and the corresponding fidelity can be achieved by maximizing MEF. max To determine, as defined by the following formula: (17); where x new Indicates updating the sample, tt new This represents the fidelity source for updating samples, with different fidelities corresponding to different MEF values. max Maximum value search is achieved using particle swarm optimization algorithm; the MEF values ​​corresponding to different fidelities are compared. max The maximum value determines the current update sample and the corresponding fidelity source. Two candidate update samples are generated from the low-fidelity model and the high-fidelity model respectively. The sample with the larger acquisition function value is selected as the update sample in step S6. Step S6-7: Based on the fidelity tt new Select the appropriate model, if tt new =h, then choose the high-fidelity model; if tt new =l, then select the low-fidelity model; update sample x new The input is fed into the corresponding model to obtain the output response; if tt new =h, then update the sample to [x new ,f hnew If tt new =h, then update the sample to [x new ,f lnew ], where f hnew x represents new The output response, f, in the high-fidelity model lnew x represents new Output response in a low-fidelity model.

6. The adaptive hierarchical Kriging model construction method based on a sequential alternating sampling strategy according to claim 1, characterized in that, In step S8, the accuracy is determined based on the maximum relative error (MAE) and the root mean square error (RMSE).

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