A variable-fidelity surrogate model construction method based on kernel regression correction factor

Abstract

The application discloses a variable-fidelity proxy model construction method based on a kernel regression correction factor, and relates to the technical field of proxy models. The method first uses low-fidelity samples to build a low-fidelity proxy model, and calculates the low-fidelity response of high-fidelity samples and a point to be predicted; then a variable-fidelity prediction model of a kernel regression correction factor is constructed, a design matrix is constructed after the high-fidelity samples are inputted, and a closed-form solution of a kernel regression coefficient is obtained by introducing ridge regression regularization; then a dimension self-adaptive scaling rule is set to generate a kernel length scale grid, and the kernel length scale and the regularization strength are automatically optimized relying on generalized cross-validation; finally, the point to be predicted is standardized and high-fidelity prediction results are calculated. The application is not constrained by input dimensions, can realize local adaptive error correction, has small prediction calculation amount, and has higher prediction accuracy and robustness in the engineering optimization scene with high dimensions and high-fidelity sample scarcity.

Description

Technical Field

[0001] This invention relates to the field of surrogate model technology, and in particular to a method for constructing a variable fidelity surrogate model based on a kernel regression correction factor. Background Technology

[0002] In the design and optimization of complex engineering systems such as aerospace, machinery, and civil engineering, high-fidelity numerical simulation can accurately reproduce the real physical operating laws of equipment and output highly reliable calculation results. However, this type of simulation suffers from extremely high computational costs; a single complete simulation calculation often takes several hours or even days. Directly using high-fidelity simulation in large-scale iterative optimization scenarios is not feasible in engineering. Surrogate model technology establishes an approximate mapping relationship between input parameters and system response using a small amount of sample data, which can significantly replace time-consuming high-precision simulation, effectively compressing the overall design cycle and reducing computational costs. However, traditional single-fidelity surrogate models have significant shortcomings: when the input dimension of the optimization problem is high, to ensure the model's prediction accuracy, a massive amount of high-fidelity samples must be collected, and the sample collection cost remains high, making it difficult to apply in most budget-constrained engineering projects. To balance modeling accuracy and computational cost, variable-fidelity surrogate modeling technology has emerged. This method integrates a small number of high-precision, high-fidelity samples with a large number of low-cost, low-fidelity samples for modeling. It relies on low-fidelity data to capture the global trend of system changes and then uses scarce high-fidelity data to complete error correction, becoming the mainstream modeling approach in the field of complex engineering optimization.

[0003] Currently, the comprehensive correction method is the most widely used variable fidelity modeling framework. The MFS-WLS method, as a representative algorithm under this framework, integrates the correction factor and the high-fidelity difference function into a weighted least squares solution. It can achieve ideal modeling results in simple cases with sufficient low-dimensional and high-fidelity samples, but it has several inherent defects that are difficult to avoid: First, the number of model parameters in this method increases linearly with the input dimension, and the number of parameters is fixed at d+2. Once the number of high-fidelity samples in actual engineering is less than d+2, the algorithm does not meet the solution conditions and cannot build an effective surrogate model. However, high-dimensional engineering problems generally suffer from a scarcity of high-fidelity samples, which can easily lead to insufficient parameters, a significant decrease in model accuracy, or even modeling failure. Second, MFS-WLS uses a global polynomial as a unified correction function, applying the same form of error correction to the entire input design space. This cannot adapt to the local nonlinear characteristics of the difference between high-fidelity and low-fidelity models as the spatial location changes, making it difficult to accurately characterize the simulation deviation in local areas.

[0004] Besides the aforementioned limitations in dimensionality and local correction, existing MFS-WLS methods also suffer from the drawback of relying on manual tuning of hyperparameters. Key hyperparameters such as kernel coefficients and polynomial order lack automated optimization mechanisms during algorithm modeling, requiring developers to rely on experience to repeatedly calculate and determine parameter combinations. This significantly increases the operational process of model building and reduces the efficiency of engineering optimization iterations. Furthermore, manual parameter tuning is highly subjective and prone to unreasonable parameter settings, directly affecting model prediction accuracy and experimental reproducibility. In summary, existing mainstream variable fidelity surrogate model methods cannot adapt to typical engineering scenarios with high dimensions and few high-fidelity samples. There is an urgent need to propose a novel variable fidelity modeling scheme to overcome the bottleneck of parameter quantity increasing with dimensionality and achieve local adaptive error correction and automatic hyperparameter optimization. Summary of the Invention

[0005] The purpose of this invention is to propose a variable fidelity surrogate model construction method based on kernel regression correction factors. This method addresses the problems of existing MFS-WLS variable fidelity surrogate models, such as the number of parameters increasing with dimensionality, the inability to achieve local adaptive correction, and the need for manual setting of hyperparameters. By introducing kernel regression correction factors to construct a variable fidelity surrogate model, this invention achieves parameters that are only related to the number of high-fidelity samples, spatial local adaptive correction, and dimensional adaptive kernel parameter grids. This method steadily improves the prediction accuracy and practicality of the model in high-dimensional, low-sample engineering scenarios.

[0006] To achieve the above objectives, this invention proposes a method for constructing a variable fidelity surrogate model based on a kernel regression correction factor, the specific steps of which are as follows: Step S1: Collect the high-fidelity sample set and low-fidelity sample set corresponding to the engineering system, construct a low-fidelity proxy model using the low-fidelity sample set, and calculate the low-fidelity response at each high-fidelity sample point and the point to be predicted using the low-fidelity proxy model. Step S2: Construct a variable fidelity prediction model based on the low-fidelity response; Step S3: Perform dimension-wise standardization on the high-fidelity sample input, construct the design matrix by combining the corresponding low-fidelity predicted values, and introduce ridge regression regularization. Solve the kernel regression coefficient vector by minimizing the sum of squared regularization errors. Step S4: Using the generalized cross-validation criterion, jointly optimize the kernel length scale and regularization intensity on the preset hyperparameter grid, and select the optimal hyperparameter combination that minimizes the generalized cross-validation value. Step S5: Set dimension adaptive scaling rules for high-dimensional scenes and generate a kernel length scale search grid that adapts to the input dimension; Step S6: Input the points to be predicted, and after standardization, combine the optimal hyperparameters to solve for the high-fidelity prediction value of the variable-fidelity model.

[0007] Preferably, in step S1, the low-fidelity surrogate model is a multi-quadratic radial basis function surrogate model, and the expression of the multi-quadratic radial basis function is as follows: ; in, For Euclidean distance, For shape parameters, leave-one-out cross-validation is used to select the candidate with the smallest error from a pre-defined candidate set. As the optimal shape parameter.

[0008] Preferably, in step S2, a variable fidelity prediction model is constructed based on the low-fidelity response, as shown in the following formula: ; in, This is the high-fidelity predicted response value output by the variable fidelity proxy model. For kernel regression correction factor, This refers to the low-fidelity predicted response value output by the low-fidelity proxy model. Design variables in the design space; Kernel regression correction factor The kernel regression form is as follows: ; in, Kernel regression coefficients For kernel function, For high-fidelity sample size, For the first i A standardized high-fidelity sample input vector.

[0009] Preferably, the kernel function is a Gaussian kernel function, and the calculation formula is as follows: ; in, The core length scale.

[0010] Preferably, in step S3, the high-fidelity sample input is standardized dimension by dimension, as shown in the following formula: ; in, For the standardized version A high-fidelity input sample, For the first i One original high-fidelity input sample, , These are the mean and standard deviation of the high-fidelity sample input, respectively. The design matrix is ​​constructed by combining the corresponding low-fidelity prediction values, as shown in the following formula: ; in, Matrix design for multi-fidelity The Line number Column elements, For the standardized version A high-fidelity input sample, This is a column vector of low-fidelity predicted values; By introducing ridge regression regularization, the kernel regression coefficient vector is solved by minimizing the sum of squared regularization errors, as shown in the following formula: ; in, For the kernel regression coefficient vector, For regularization strength, It is the identity matrix. Matrix design for multi-fidelity For designing a matrix The transpose of the matrix, This is the vector of true response values ​​for high-fidelity samples.

[0011] Preferably, in step S4, the generalized cross-validation formula is as follows: ; in, For generalized cross-validation functions, This represents the total number of samples participating in the training. For the hat matrix, The trace of the hat matrix; Hat Matrix The formula is as follows: ; By using the preset × A grid search is performed to find the combination that minimizes the GCV. .

[0012] Preferably, in step S5, the standardized Euclidean distance between points in the high-dimensional space is calculated according to... Growth, nuclear length scale The search grid scales adaptively by dimension, as shown in the following formula: ; ; This is a scaling factor used to dynamically adjust the search range of hyperparameters based on the input dimension; For adaptive The search grid is the set of candidate values ​​for which the model actually performs hyperparameter optimization in the current dimension; The base kernel length scale grid serves as the preset initial hyperparameter search space; For input dimensions; when When the value is ≤6, the scaling factor is 1, which does not change the search range; when When the value is greater than 6, the search range will be automatically expanded.

[0013] Preferably, in step S6, the prediction process for the point to be predicted is as follows: Step S61: First, determine the points to be predicted. Standardization is achieved The Gaussian kernel vector between the predicted point and all high-fidelity standardized samples is calculated using the following formula: ; in, Points to be predicted The kernel function value vector between all high-fidelity training sample points, For the standardized version A high-fidelity input sample; The final high-fidelity prediction value is calculated using the following formula: ; in, For multi-fidelity surrogate models at the point to be predicted High-fidelity predicted response value at the location, For low-fidelity surrogate models at the point to be predicted Low-fidelity predicted response value at the location.

[0014] Therefore, this invention proposes a method for constructing a variable fidelity surrogate model based on a kernel regression correction factor, the advantages of which are as follows: (1) This invention eliminates the defect that the number of parameters in the traditional MFS-WLS method increases linearly with the input dimension. The number of model parameters is only equal to the number of high-fidelity samples and is not constrained by the input dimension. In engineering scenarios where high-dimensional, high-fidelity samples are scarce and the number of samples is less than the design dimension, there will be no modeling failure, which greatly improves the applicability and robustness of the model.

[0015] (2) This invention uses Gaussian kernel regression to construct a position-related correction factor to replace the global unified polynomial correction. The correction magnitude can be dynamically adjusted according to the magnitude of the high-fidelity simulation deviation in different regions of the design space, accurately capturing nonlinear local differences. Compared with the global correction method, it significantly reduces the model prediction error.

[0016] (3) The present invention designs a dimension-adaptive kernel length scale scaling mesh, which automatically widens the kernel parameter search range to address the Euclidean distance degradation problem in high-dimensional space, ensuring that the Gaussian kernel still has effective spatial differentiation and local perception capabilities under high-dimensional input, thus avoiding a significant decline in the accuracy of high-dimensional modeling.

[0017] (4) This invention uses generalized cross-validation (GCV) as the target criterion and automatically solves the optimal kernel length scale and ridge regression regularization strength by traversing the grid. It eliminates the need to manually adjust hyperparameters based on experience, simplifies the modeling process, and eliminates the subjective error caused by manual parameter tuning, thereby improving the reproducibility of the results.

[0018] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0019] Figure 1 This is a flowchart of a method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to the present invention; Figure 2 This is a model diagram of a variable fidelity surrogate model construction method based on kernel regression correction factor according to the present invention; Figure 3 This invention addresses various methods R under the condition of equal cost and low dimension. 2 Performance difference comparison chart; Figure 4 This is a comparison chart of the NMAE performance differences of various methods under the same cost and low dimension conditions. Figure 5 This invention addresses various methods R under the condition of equal cost and high dimensionality. 2 Performance difference comparison chart; Figure 6 This is a comparison chart of the NMAE performance differences of various methods under the same cost and high dimensionality conditions. Figure 7 For the verification of the multi-fidelity value of this invention R 2 Result comparison chart; Figure 8 This is a comparison chart of the NMAE results for the multifidelity verification of this invention; Figure 9 The cost-to-sensitivity ratio and MFS-KR performance of each method in this invention are shown in Figure R. 2 Result comparison chart; Figure 10 This is a comparison chart of the cost-to-sensitivity ratio and MFS-KR performance NMAE results of various methods in this invention. Detailed Implementation

[0020] To make the technical solutions, advantages, and objectives of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below. The described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0021] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0022] Example 1 like Figures 1-2 As shown, this invention provides a method for constructing a variable fidelity surrogate model based on a kernel regression correction factor (MFS-KR method), the specific steps of which are as follows: Step S1: Collect the high-fidelity sample set and low-fidelity sample set corresponding to the engineering system, construct a low-fidelity proxy model using the low-fidelity sample set, and calculate the low-fidelity response at each high-fidelity sample point and the point to be predicted using the low-fidelity proxy model. Among them, the low-fidelity surrogate model is a multi-quadratic radial basis function surrogate model, and the expression of the multi-quadratic radial basis function is as follows: ; in, For Euclidean distance, For shape parameters, leave-one-out cross-validation is used to select the candidate with the smallest error from a pre-defined candidate set. As the optimal shape parameter.

[0023] Step S2: Construct a variable fidelity prediction model based on the low-fidelity response, as shown in the following formula: ; in, This is the high-fidelity predicted response value output by the variable fidelity proxy model. For kernel regression correction factor, This refers to the low-fidelity predicted response value output by the low-fidelity proxy model. Design variables in the design space; Kernel regression correction factor The kernel regression form is as follows: ; in, Kernel regression coefficients For kernel function, For high-fidelity sample size, For the first i A standardized high-fidelity sample input vector.

[0024] The kernel function is the Gaussian kernel function, and the calculation formula is as follows: ; in, The core length scale.

[0025] Step S3: Perform dimension-wise standardization on the high-fidelity sample input, construct the design matrix by combining the corresponding low-fidelity predicted values, and introduce ridge regression regularization. Solve the kernel regression coefficient vector by minimizing the sum of squared regularization errors. The high-fidelity sample input is standardized dimension-by-dimensionally, using the following formula: ; in, For the standardized version A high-fidelity input sample, For the first i One original high-fidelity input sample, , These are the mean and standard deviation of the high-fidelity sample input, respectively. The design matrix is ​​constructed by combining the corresponding low-fidelity prediction values, as shown in the following formula: ; in, Matrix design for multi-fidelity The Line number Column elements, For the standardized version A high-fidelity input sample, This is a column vector of low-fidelity predicted values; By introducing ridge regression regularization, the kernel regression coefficient vector is solved by minimizing the sum of squared regularization errors, as shown in the following formula: ; in, For the kernel regression coefficient vector, For regularization strength, It is the identity matrix. Matrix design for multi-fidelity For designing a matrix The transpose of the matrix, This is the vector of true response values ​​for high-fidelity samples.

[0026] Step S4: Using the generalized cross-validation criterion, jointly optimize the kernel length scale and regularization intensity on the preset hyperparameter grid, and select the optimal hyperparameter combination that minimizes the generalized cross-validation value. The generalized cross-validation formula is as follows: ; in, For generalized cross-validation functions, This represents the total number of samples participating in the training. For the hat matrix, The trace of the hat matrix; Hat Matrix The formula is as follows: ; By using the preset × A grid search is performed to find the combination that minimizes the GCV. .

[0027] Step S5: Set dimension adaptive scaling rules for high-dimensional scenes and generate a kernel length scale search grid that adapts to the input dimension; Standardized Euclidean distance between points in high-dimensional space Growth, nuclear length scale The search grid scales adaptively by dimension, as shown in the following formula: ; ; in, This is a scaling factor used to dynamically adjust the search range of hyperparameters based on the input dimension; For adaptive The search grid is the set of candidate values ​​for which the model actually performs hyperparameter optimization in the current dimension; The base kernel length scale grid serves as the preset initial hyperparameter search space; For input dimensions; when When the value is ≤6, the scaling factor is 1, which does not change the search range; when When the value is greater than 6, the search range will be automatically expanded.

[0028] Step S6: Input the points to be predicted, and after standardization, combine the optimal hyperparameters to solve for the high-fidelity prediction value of the variable-fidelity model.

[0029] The specific prediction process for the points to be predicted is as follows: Step S61: First, determine the points to be predicted. Standardization is achieved The Gaussian kernel vector between the predicted point and all high-fidelity standardized samples is calculated using the following formula: ; in, Points to be predicted The kernel function value vector between all high-fidelity training sample points, For the standardized version A high-fidelity input sample; The final high-fidelity prediction value is calculated using the following formula: ; in, For multi-fidelity surrogate models at the point to be predicted High-fidelity predicted response value at the location, For low-fidelity surrogate models at the point to be predicted Low-fidelity predicted response value at the location.

[0030] To test the actual performance of this invention, this embodiment selects eight standard test functions: Forrester (1D), Branin (2D), Park91a (4D), Hartmann6 (6D), Borehole (8D), Dixon-Price (10D), and Styblinski-Tang (12D / 16D), covering dimensions from 1D to 16D. A cost-comparison experiment is conducted with the MFS-WLS method, the AR1-type Co-Kriging method, and the full high-fidelity RBF method. The computational cost of evaluating a high-fidelity sample is assumed to be δ times that of evaluating a low-fidelity sample. With a fixed total budget, each configuration is independently sampled 20 times and the average is taken. The coefficient of determination R0 is used. 2 The mean absolute error (NMAE) was used as the evaluation metric. Error bars were introduced into the results charts to quantify the uncertainty of the model performance. The center point of the error bar represents the R-value of 20 independent repeated experiments. 2 The mean and upper and lower limits correspond to the 95% confidence interval (the reliable range of model performance), which are used to evaluate the reliable range and stability of the model's expected performance.

[0031] like Figures 3-6As shown, for low-dimensional test conditions, the traditional MFS-WLS method requires fewer model parameters, and the number of high-fidelity samples is sufficient to meet its solution conditions. Furthermore, the global polynomial correction form can effectively characterize the mapping differences between high-fidelity and low-fidelity simulations. Therefore, MFS-WLS achieves excellent prediction accuracy under these conditions. The MFS-KR method of this invention relies on kernel regression to construct a local adaptive correction term, resulting in slightly higher model complexity and slightly lower prediction accuracy than MFS-WLS under simple low-dimensional conditions. When the input dimension of the test function increases and the nonlinear characteristics of the system strengthen, the performance of the MFS-WLS method significantly degrades. The prediction performance of the MFS-KR method of this invention is comprehensively superior to that of the MFS-WLS method. Especially in modeling scenarios where high-fidelity samples are scarce, the number of parameters required by the MFS-WLS method increases linearly with the input dimension. A limited number of high-fidelity samples cannot support complete parameter solving, making it difficult to accurately characterize the complex bias relationship between high-fidelity and low-fidelity models, resulting in a significant deterioration in prediction accuracy. In extreme cases, it may even be impossible to construct an effective surrogate model. In contrast, the MFS-KR method of this invention has a model parameter count that is consistent only with the number of high-fidelity samples, is not constrained by the input dimension, and can achieve stable modeling under any dimension and any number of samples, with a model determination coefficient R0. 2 It consistently remained in a highly stable range, fully verifying that the kernel regression local correction architecture possesses extremely strong modeling robustness.

[0032] like Figures 7-8 As shown, for high-dimensional numerical examples, the single-fidelity modeling baseline using only 2d high-fidelity samples has obvious defects. The number of samples is severely insufficient, and the prediction accuracy of this baseline scheme is extremely poor in most cases, with the coefficient of determination R0 being significantly lower. 2 Negative values ​​can occur, and under extreme conditions, it may even be impossible to construct a surrogate model with effective predictive capabilities. The MFS-KR method of this invention relies on only a small number of high-fidelity samples, combined with a large batch of low-cost low-fidelity samples, to complete modeling. The resulting prediction accuracy is significantly improved compared to a single-fidelity baseline, with small overall performance fluctuations and excellent modeling robustness. In contrast, the traditional MFS-WLS method exhibits large fluctuations in prediction accuracy under different test functions. When high-fidelity samples are scarce, it is prone to modeling failure, failing to construct an effective surrogate model. The number of parameters required by the MFS-WLS method increases linearly with the input dimension, and a small number of high-fidelity samples are insufficient to meet its parameter solution conditions, making it unable to accurately represent the deviation mapping relationship between high-fidelity and low-fidelity simulations. The MFS-KR method of this invention, however, has a parameter count determined only by the number of high-fidelity samples, is not limited by the input dimension, and can meet the parameter solution requirements under any working condition, demonstrating stronger adaptability to high-dimensional, low-sample engineering scenarios.

[0033] Figures 9-10 To compare performance curves under different cost ratios δ, by Figures 9-10It can be seen that as the cost ratio δ increases, the number of low-fidelity samples available for modeling increases accordingly, and the fitting accuracy of the low-fidelity surrogate model to the global trend of the system improves. The predictive performance of various variable-fidelity modeling methods is also improved. Within the cost ratio range of all tests, the overall performance of the MFS-KR method of this invention is better than the comparative algorithms; and when the cost ratio δ is small, the performance advantage of MFS-KR is more prominent. Under this condition, the stock of low-fidelity samples is limited, and the fitting error of the low-fidelity surrogate model itself is large. MFS-WLS, which uses global polynomial correction, is unable to compensate for local spatial biases, resulting in poor reliability of the correction effect. In contrast, MFS-KR relies on kernel regression to achieve a local adaptive correction mechanism, which can fully explore the trend information contained in the limited low-fidelity samples and more accurately restore the overall change law of the system. Observing the performance curves of the MFS-KR method under various cost ratios, it can be seen that the five curves are hierarchically ordered. The larger the cost ratio δ, the higher the predictive performance index of the corresponding curve, proving that this invention can make full use of the data value of all low-fidelity samples, maintain stable prediction accuracy under different simulation cost ratios, and has good modeling robustness.

[0034] It is worth noting that all contents not described in detail in this invention are existing technologies and are well known to those skilled in the art.

[0035] Therefore, this invention provides a method for constructing a variable-fidelity surrogate model based on a kernel regression correction factor. It adopts a kernel regression local adaptive correction factor, and the number of model parameters is determined only by the number of high-fidelity samples, which solves the defects of the traditional MFS-WLS method where the number of parameters increases with the dimension and high-dimensional few samples are prone to modeling failure. At the same time, it is combined with a dimension-adaptive kernel parameter grid and GCV automatic hyperparameter optimization, which takes into account both the high-dimensional space fitting ability and the modeling convenience. It has stable prediction accuracy and excellent robustness under various sample ratios and simulation cost ratios, and is suitable for high-dimensional few-sample engineering optimization scenarios.

[0036] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for constructing a variable fidelity surrogate model based on a kernel regression correction factor, characterized in that, The specific steps are as follows: Step S1: Collect the high-fidelity sample set and low-fidelity sample set corresponding to the engineering system, construct a low-fidelity proxy model using the low-fidelity sample set, and calculate the low-fidelity response at each high-fidelity sample point and the point to be predicted using the low-fidelity proxy model. Step S2: Construct a variable fidelity prediction model based on the low-fidelity response; Step S3: Perform dimension-wise standardization on the high-fidelity sample input, construct the design matrix by combining the corresponding low-fidelity predicted values, and introduce ridge regression regularization. Solve the kernel regression coefficient vector by minimizing the sum of squared regularization errors. Step S4: Using the generalized cross-validation criterion, jointly optimize the kernel length scale and regularization intensity on the preset hyperparameter grid, and select the optimal hyperparameter combination that minimizes the generalized cross-validation value. Step S5: Set dimension adaptive scaling rules for high-dimensional scenes and generate a kernel length scale search grid that adapts to the input dimension; Step S6: Input the points to be predicted, and after standardization, combine the optimal hyperparameters to solve for the high-fidelity prediction value of the variable-fidelity model.

2. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 1, characterized in that, In step S1, the low-fidelity surrogate model is a multi-quadratic radial basis function surrogate model, and the expression of the multi-quadratic radial basis function is as follows: ； in, For Euclidean distance, For shape parameters, leave-one-out cross-validation is used to select the candidate with the smallest error from a pre-defined candidate set. As the optimal shape parameter.

3. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 2, characterized in that, In step S2, a variable fidelity prediction model is constructed based on the low-fidelity response, as shown in the following formula: ； in, This is the high-fidelity predicted response value output by the variable fidelity proxy model. For kernel regression correction factor, This refers to the low-fidelity predicted response value output by the low-fidelity proxy model. Design variables in the design space; Kernel regression correction factor The kernel regression form is as follows: ； in, Kernel regression coefficients For kernel function, For high-fidelity sample size, For the first i A standardized high-fidelity sample input vector.

4. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 3, characterized in that, The kernel function is a Gaussian kernel function, and the calculation formula is as follows: ； in, The core length scale.

5. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 4, characterized in that, In step S3, the high-fidelity sample input is standardized dimension by dimension, as shown in the following formula: ； in, For the standardized version A high-fidelity input sample, For the first i One original high-fidelity input sample, , These are the mean and standard deviation of the high-fidelity sample input, respectively. The design matrix is ​​constructed by combining the corresponding low-fidelity prediction values, as shown in the following formula: ； in, Matrix design for multi-fidelity The Line number Column elements, For the standardized version A high-fidelity input sample, This is a column vector of low-fidelity predicted values; By introducing ridge regression regularization, the kernel regression coefficient vector is solved by minimizing the sum of squared regularization errors, as shown in the following formula: ； in, For the kernel regression coefficient vector, For regularization strength, It is the identity matrix. Matrix design for multi-fidelity For designing a matrix The transpose of the matrix, This is the vector of true response values ​​for high-fidelity samples.

6. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 5, characterized in that, In step S4, the generalized cross-validation formula is as follows: ； in, For generalized cross-validation functions, This represents the total number of samples participating in the training. For the hat matrix, The trace of the hat matrix; Hat Matrix The formula is as follows: ； By using the preset × A grid search is performed to find the combination that minimizes the GCV. .

7. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 6, characterized in that, In step S5, the standardized Euclidean distance between points in the high-dimensional space is calculated according to... Growth, nuclear length scale The search grid scales adaptively by dimension, as shown in the following formula: ； ； in, Scaling factor For adaptive Search grid, Based on the kernel length scale grid For input dimensions; when When the value is ≤6, the scaling factor is 1, which does not change the search range; when When the value is greater than 6, the search range will be automatically expanded.

8. The method for constructing a variable fidelity surrogate model based on a kernel regression correction factor according to claim 7, characterized in that, In step S6, the prediction process for the points to be predicted is as follows: Step S61: First, determine the points to be predicted. Standardization is achieved The Gaussian kernel vector between the predicted point and all high-fidelity standardized samples is calculated using the following formula: ； in, Points to be predicted The kernel function value vector between all high-fidelity training sample points, For the standardized version A high-fidelity input sample; The final high-fidelity prediction value is calculated using the following formula: ； in, For multi-fidelity surrogate models at the point to be predicted High-fidelity predicted response value at the location, For low-fidelity surrogate models at the point to be predicted Low-fidelity predicted response value at the location.

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