Multi-fidelity model construction method based on hyperplane correction
By learning the feature mapping between high-fidelity and low-fidelity models through the hyperplane correction mechanism, a multi-fidelity surrogate model is constructed, which solves the problem of insufficient capture of nonlinear deviation between high-fidelity and low-fidelity models in the existing technology, and achieves higher accuracy and robust prediction.
Patent Information
- Application Number
- CN202610140861.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-02-02
- Publication Date
- 2026-05-01
AI Technical Summary
Existing multifidelity model building techniques cannot effectively capture and correct the complex spatial nonlinear deviations between high-fidelity and low-fidelity models, resulting in insufficient prediction accuracy and generalization ability in complex engineering optimization problems.
By employing a hyperplane correction mechanism, a multi-fidelity proxy model is constructed by learning the feature mapping relationship between high-fidelity and low-fidelity models, which significantly reduces the dependence on the number of high-fidelity samples and achieves accurate correction.
The constructed multi-fidelity model exhibits better generalization performance and prediction robustness when facing nonlinear and multimodal problems, reduces the risk of optimization decision errors, and provides quantitative basis for adjusting the predictions of the low-fidelity model.
Smart Images
Figure CN121959957A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of multifidelity proxy model technology, specifically relating to a method for constructing a multifidelity model based on hyperplane correction. Background Technology
[0002] In engineering design and scientific computing fields, such as the analysis and optimization of complex systems in aerospace, vehicle engineering, and electronic equipment, engineers generally rely on high-fidelity simulation models to obtain accurate analysis results. However, such models are often extremely computationally intensive, with a single simulation potentially taking hours or even days, making it difficult to implement optimization designs based on numerous iterations, uncertainty quantification, or parameter scanning. To address this challenge, multi-fidelity modeling techniques have emerged. The core idea of this technique is to integrate computationally inexpensive but limited-accuracy low-fidelity models (such as simplified physical models, coarse-mesh models, or empirical models) with computationally expensive high-fidelity models. The aim is to utilize a small number of high-fidelity samples and a large number of low-fidelity samples to construct a surrogate model that combines high accuracy and high efficiency, thereby significantly reducing the overall computational cost.
[0003] Currently, one of the core ideas in multi-fidelity model construction methods is to establish the correlation and correction relationship between the outputs of high-fidelity and low-fidelity models. Classical methods (such as Co-Kriging models) typically assume a global linear or stationary correlation between the high-fidelity and low-fidelity models, approximating the high-fidelity model by calibrating the scalar and offset of the low-fidelity model. However, for complex engineering problems with highly nonlinear and multimodal characteristics, this global linear or stationary assumption is often oversimplified, leading to a significant decrease in the prediction accuracy of the fusion model in unsampled regions and insufficient generalization ability. In summary, the core dilemma facing current multi-fidelity model construction technology is the inability to simultaneously and efficiently capture and correct the complex spatial nonlinear biases and key feature phase biases between the low-fidelity and high-fidelity models. This severely restricts the reliable application and accuracy ceiling of multi-fidelity technology in complex engineering optimization problems. Summary of the Invention
[0004] To address the shortcomings of existing technologies, this invention proposes a multi-fidelity model construction method based on hyperplane correction, which approximates the expensive high-fidelity model using a multi-fidelity model. The core of this method lies in introducing a "hyperplane correction" mechanism, which explicitly learns and corrects the feature mapping relationships between models of different fidelities in the input space. This significantly reduces the dependence on the number of high-fidelity samples while achieving accurate correction of both low-fidelity and high-fidelity models, ultimately constructing a multi-fidelity surrogate model with high prediction accuracy and strong generalization ability.
[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for constructing a multi-fidelity model based on hyperplane correction. The method includes the following steps: Step S1: Obtain the corresponding high-fidelity model within the existing modeling and simulation environment. g e ( x Low-fidelity model g l ( x And the sample space of the input variables. High-fidelity input samples are obtained using Latin hypercube experimental design. x e and low-fidelity input samples x l In addition, high-fidelity input validation samples were obtained using Latin hypercube experimental design. x t .
[0006] Step S2: Process high-fidelity input samples x e Obtain the output response value corresponding to the high-fidelity model Y e For low-fidelity input samples x l Obtain the output response value corresponding to the low-fidelity model Y l This leads to a high-fidelity training sample dataset. x e , Y e ] and low-fidelity training sample data set[ x l , Y l Additionally, high-fidelity input verification samples were used. x t Obtain the output response value corresponding to the high-fidelity model Y t A high-fidelity verification sample dataset was obtained. x t , Y t ].
[0007] Step S3: Based on the low-fidelity training sample dataset [ x l , Y l Use the Kriging model to build a low-fidelity model.
[0008] Step S3-1: According to Kriging theory, a low-fidelity model can be expressed as the sum of a polynomial and a random distribution function: (1) In the formula, This represents a low-fidelity model. x Indicates the input prediction sample; Represents the regression parameters of the low-fidelity model; This represents the regression term of the low-fidelity model; Let Kriging represent a random distribution function, i.e., the prediction uncertainty introduced by the Kriging model, whose distribution is: .in, Indicates a normal distribution. This represents the mean squared error of the Kriging model's predictions.
[0009] Step S3-2: The obtained low-fidelity model follows a normal distribution, and its prediction mean is... y l .
[0010] Step S4: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ], establish the predicted mean of the hyperplane model y line The specific expression: (2) In the formula, e Represents a unit vector; y l Indicates the predicted sample x The corresponding low-fidelity model prediction mean; y line This represents the mean of the hyperplane model predictions for the predicted sample.
[0011] Step S5: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ],as well as y line and y l The specific expression for the predicted mean of a multifidelity model is defined as follows: (3) In the formula, This represents the predicted mean of the hyperplane model; This represents the regression parameters corresponding to the hyperplane model; This represents the error term in a multifidelity model.
[0012] Step S5-1: The error term of the multifidelity model is represented by a stochastic process with the following distribution: (4) In the formula, This represents the mean squared error of the predictions in a multifidelity model. , and Perform splicing, define Represented as , This represents a concatenated sample matrix.
[0013] Step S5-2: Integrate all regression terms They are defined as follows: (5) In the formula, w This represents the predicted mean of the hyperplane model; v This represents the regression parameters corresponding to the hyperplane model.
[0014] Step S5-3: For A high-fidelity training sample, regression term matrix It is expressed as follows: (6) In the formula, Indicates the first One training sample; This represents the mean of the hyperplane model predictions for the first training sample; This represents the mean of the hyperplane model predictions for the second training sample; Indicates the first The mean of hyperplane model predictions for each training sample. W The corresponding training sample error term is represented as: ,in This represents the error term of the hyperplane model for the first training sample; This represents the hyperplane model error term for the second training sample; Indicates the first Error term of the hyperplane model for each training sample.
[0015] Step S5-4: The covariance function of the correlation coefficient of the error terms for different training samples is expressed as follows: (7) In the formula, Indicates process variance; Hyperparameters Related functions; Represents the covariance function; a This represents the first training sample in the correlation function; b This represents the second training sample in the correlation function; express a Error term of the hyperplane model for training samples; express b Error term of the hyperplane model for training samples; This represents the expectation function. Specifically, it is derived from the first... i A spliced sample matrix of high-fidelity training samples and the j A spliced sample matrix of high-fidelity training samples The mathematical expression for the correlation function of the sample composition is: ,in n Indicates the sample dimension; k Indicates the dimension number; Indicates the first k Hyperparameters corresponding to each sample dimension; Indicates the first i The first concatenated sample matrix in the nth concatenated sample matrix k One dimension; Indicates the first j The first concatenated sample matrix in the nth concatenated sample matrix k Each dimension.
[0016] Step S5-5: Predict the sample x Correlation function between and high-fidelity training samples Represented as: (8) In the formula, This represents the concatenated sample matrix of the first high-fidelity training sample; This represents the concatenated sample matrix of the second high-fidelity training sample; Indicates the first A spliced sample matrix of high-fidelity training samples.
[0017] Steps S5-6: For ease of description, the predicted values of the predicted samples are treated as a linear combination of the observed samples, and the estimated mean value of the multifidelity model prediction is further expressed as: (9) In the formula, This represents the mean estimate of the multifidelity model prediction; This represents the output response value corresponding to the high-fidelity model. The corresponding weighting factor matrix; This represents the first weighting factor; This represents the second weighting factor; Indicates the first Each weighted factor.
[0018] Step S5-7: The prediction error of the established multi-fidelity model is expressed as: (10) In the formula, This represents the output response value corresponding to the high-fidelity model. The corresponding weighting factor matrix; This represents the training sample error term of the multifidelity model; This represents the prediction sample error term in the multifidelity model.
[0019] Step S5-8: Under the condition of unbiased estimation, let Mean square error of multifidelity model Represented as: (11) Step S5-9: In order to calculate c The Lagrange function can be constructed as follows: (12) In the formula, λ It represents the Lagrange multiplier.
[0020] Step S5-10: Solve the Lagrange linear equations. The final parameter results are as follows: (13) Step S5-11: The estimated value of the predicted mean of the established multi-fidelity model is: (14) In the formula, This represents an estimate of the predicted mean of the multifidelity model. , .
[0021] Mean squared error of prediction in multifidelity models The expression is: (15) In the formula, Process variance Solve using maximum likelihood estimation.
[0022] Step S6: Solve for the hyperparameters of the established multifidelity model using maximum likelihood estimation.
[0023] Step S6-1: Define the joint probability density function of all high-fidelity training samples as follows: (16) In the formula, This represents the joint probability density function of high-fidelity training samples. This represents hyperparameters.
[0024] Step S6-2: Treat the joint probability density function as a likelihood function and perform a maximization operation, respectively for... and Taking the partial derivatives, we can solve for: (17) Substituting the result into the likelihood function, while ignoring the constant term, we further present: (18) This invention uses a pattern search algorithm to calculate the optimal value of hyperparameters.
[0025] Step S7: Based on the model hyperparameters, obtain the process variance calculation results, further obtain the estimated value of the predicted mean and the predicted root mean square error of the multi-fidelity model, and output the final multi-fidelity surrogate model based on hyperplane correction. Finally, perform accuracy prediction on the high-fidelity validation samples to obtain the error evaluation results of maximum relative error (MAE) and root mean square error (RMSE).
[0026] The beneficial effects of this invention are: This invention, by incorporating a hyperplane correction mechanism, can more fundamentally capture the intrinsic mapping patterns between models of different fidelity levels, resulting in a final surrogate model with better adaptability and stability to changes in the input space. When facing new design points or operating conditions with strong nonlinearity and multimodal characteristics, the model constructed by this invention exhibits superior generalization performance and prediction robustness, reducing the risk of optimization decision errors due to model mismatch. Furthermore, the parameters in the hyperplane correction mechanism itself contain quantitative information on how the predicted values of the low-fidelity model should be adjusted in different input regions to approximate the high-fidelity response. This provides an intuitive quantitative basis for the main sources of differences between high-fidelity and low-fidelity models, offering better interpretability. Attached Figure Description
[0027] Figure 1 This is a flowchart of an embodiment of the present invention; Figure 2 This is a prediction result diagram of Embodiment 1 of the present invention; Figure 3 This is a box plot of the prediction error from Embodiment 2 of the present invention; Figure 3 (a) in the figure is a box plot comparing the MAE error results; Figure 3(b) in the figure is a box plot comparing the RMSE error results. Detailed Implementation
[0028] 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.
[0029] Example 1: As Figure 1 As shown, this embodiment provides a method for constructing a multi-fidelity model based on hyperplane correction, including the following steps: Step S1: Obtain the corresponding high-fidelity model within the existing modeling and simulation environment. g e ( x Low-fidelity model g l ( x The sample space of the 1-dimensional input variables is [0, 10]. Five high-fidelity input samples were obtained using a Latin hypercube experimental design. x e and 10 low-fidelity input samples x l In addition, high-fidelity input validation samples were obtained using Latin hypercube experimental design. x t Among them, the high-fidelity model g e ( x Low-fidelity model g l ( x The expressions for ) are as follows: (19) Step S2: Process high-fidelity input samples x e Obtain the output response value corresponding to the high-fidelity model Y e For low-fidelity input samples x l Obtain the output response value corresponding to the low-fidelity model Y l This leads to a high-fidelity training sample dataset. x e , Y e ] and low-fidelity training sample data set[ x l , Y l Additionally, high-fidelity input verification samples were used. x t Obtain the output response value corresponding to the high-fidelity modelY t A high-fidelity verification sample dataset was obtained. x t , Y t ].
[0030] Step S3: Based on the low-fidelity training sample dataset [ x l , Y l Use the Kriging model to build a low-fidelity model.
[0031] Step S3-1: According to Kriging theory, a low-fidelity model can be expressed as the sum of a polynomial and a random distribution function: (1) In the formula, This represents a low-fidelity model. x Indicates the input prediction sample; Represents the regression parameters of the low-fidelity model; This represents the regression term of the low-fidelity model; Let Kriging represent a random distribution function, i.e., the prediction uncertainty introduced by the Kriging model, whose distribution is: .in, Indicates a normal distribution. This represents the mean squared error of the Kriging model's predictions.
[0032] Step S3-2: The obtained low-fidelity model follows a normal distribution, and its prediction mean is... y l .
[0033] Step S4: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ], establish the predicted mean of the hyperplane model y line The specific expression: (2) In the formula, e Represents a unit vector; y l Indicates the predicted sample x The corresponding low-fidelity model prediction mean; y lineThis represents the mean of the hyperplane model predictions for the predicted sample.
[0034] Step S5: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ],as well as y line and y l The specific expression for the predicted mean of a multifidelity model is defined as follows: (3) In the formula, This represents the predicted mean of the hyperplane model; This represents the regression parameters corresponding to the hyperplane model; This represents the error term in a multifidelity model.
[0035] Step S5-1: The error term of the multifidelity model is represented by a stochastic process with the following distribution: (4) In the formula, This represents the mean squared error of the predictions in a multifidelity model. , and Perform splicing, define Represented as , This represents a concatenated sample matrix.
[0036] Step S5-2: Integrate all regression terms They are defined as follows: (5) In the formula, w This represents the predicted mean of the hyperplane model; v This represents the regression parameters corresponding to the hyperplane model.
[0037] Step S5-3: For A high-fidelity training sample, regression term matrix It is expressed as follows: (6) In the formula, Indicates the first One training sample; This represents the mean of the hyperplane model predictions for the first training sample; This represents the mean of the hyperplane model predictions for the second training sample; Indicates the first The mean of hyperplane model predictions for each training sample. W The corresponding training sample error term is represented as: ,in This represents the error term of the hyperplane model for the first training sample; This represents the hyperplane model error term for the second training sample; Indicates the first Error term of the hyperplane model for each training sample.
[0038] Step S5-4: The covariance function of the correlation coefficient of the error terms for different training samples is expressed as follows: (7) In the formula, Indicates process variance; Hyperparameters Related functions; Represents the covariance function; a This represents the first training sample in the correlation function; b This represents the second training sample in the correlation function; express a Error term of the hyperplane model for training samples; express b Error term of the hyperplane model for training samples; This represents the expectation function. Specifically, it is derived from the first... i A spliced sample matrix of high-fidelity training samples and the j A spliced sample matrix of high-fidelity training samples The mathematical expression for the correlation function of the sample composition is: ,in n Indicates the sample dimension; k Indicates the dimension number; Indicates the first k Hyperparameters corresponding to each sample dimension; Indicates the first i The first concatenated sample matrix in the nth concatenated sample matrix k One dimension; Indicates the first j The first concatenated sample matrix in the nth concatenated sample matrix k Each dimension.
[0039] Step S5-5: Predict the sample x Correlation function between and high-fidelity training samples Represented as: (8) In the formula, This represents the concatenated sample matrix of the first high-fidelity training sample; This represents the concatenated sample matrix of the second high-fidelity training sample; Indicates the first A spliced sample matrix of high-fidelity training samples.
[0040] Steps S5-6: For ease of description, the predicted values of the predicted samples are treated as a linear combination of the observed samples, and the estimated mean value of the multifidelity model prediction is further expressed as: (9) In the formula, This represents the mean estimate of the multifidelity model prediction; This represents the output response value corresponding to the high-fidelity model. The corresponding weighting factor matrix; This represents the first weighting factor; This represents the second weighting factor; Indicates the first Each weighted factor.
[0041] Step S5-7: The prediction error of the established multi-fidelity model is expressed as: (10) In the formula, This represents the output response value corresponding to the high-fidelity model. The corresponding weighting factor matrix; This represents the training sample error term of the multifidelity model; This represents the prediction sample error term in the multifidelity model.
[0042] Step S5-8: Under the condition of unbiased estimation, let Mean square error of multifidelity model Represented as: (11) Step S5-9: In order to calculate c The Lagrange function can be constructed as follows: (12) In the formula, λ It represents the Lagrange multiplier.
[0043] Step S5-10: Solve the Lagrange linear equations. The final parameter results are as follows: (13) Step S5-11: The estimated value of the predicted mean of the established multi-fidelity model is: (14) In the formula, This represents an estimate of the predicted mean of the multifidelity model. , .
[0044] Mean squared error of prediction in multifidelity models The expression is: (15) In the formula, Process variance Solve using maximum likelihood estimation.
[0045] Step S6: Solve for the hyperparameters of the established multifidelity model using maximum likelihood estimation.
[0046] Step S6-1: Define the joint probability density function of all high-fidelity training samples as follows: (16) In the formula, This represents the joint probability density function of high-fidelity training samples. This represents hyperparameters.
[0047] Step S6-2: Treat the joint probability density function as a likelihood function and perform a maximization operation, respectively for... and Taking the partial derivatives, we can solve for: (17) Substituting the result into the likelihood function, while ignoring the constant term, we further present: (18) This invention uses a pattern search algorithm to calculate the optimal value of hyperparameters.
[0048] Step S7: Based on the process variance calculation results obtained from the model hyperparameters, further obtain the estimated value of the predicted mean and the predicted root mean square error of the multi-fidelity model, and output the final multi-fidelity surrogate model based on hyperplane correction. Finally, perform accuracy prediction on the high-fidelity validation samples to obtain the error evaluation results of maximum relative error (MAE) and root mean square error (RMSE). The method of this invention is compared with the Co-Kriging model, Co-RBF model, HK model, and IHK model. All methods use the same high-fidelity training sample dataset, low-fidelity training sample dataset, and high-fidelity validation sample dataset. The obtained high-fidelity model prediction results are as follows: Figure 2As shown in the figure, the prediction result of the method of this invention is closest to the high-fidelity true function image, indicating that the prediction accuracy of this invention is superior to the other four methods. Therefore, the multi-fidelity model construction method based on hyperplane correction proposed in this invention can obtain more accurate prediction results.
[0049] Example 2: Figure 1 As shown, this embodiment provides a method for constructing a multi-fidelity model based on hyperplane correction, including the following steps: Step S1: Obtain the corresponding high-fidelity model within the existing modeling and simulation environment. g e ( x Low-fidelity model g l ( x The sample space of the 10-dimensional input variables is [0, 10]. One hundred high-fidelity input samples were obtained using a Latin hypercube experimental design. x e and 200 low-fidelity input samples x l In addition, 1000 high-fidelity validation samples were obtained using a Latin hypercube experimental design. x t Used to verify model accuracy. High-fidelity model. g e Low-fidelity model g l The sample spaces are as follows: (20) Step S2: Process high-fidelity input samples x e Obtain the output response value corresponding to the high-fidelity model Y e For low-fidelity input samples x l Obtain the output response value corresponding to the low-fidelity model Y l This leads to a high-fidelity training sample dataset. x e , Y e ] and low-fidelity training sample data set[ x l , Y l Additionally, high-fidelity input verification samples were used. x t Obtain the output response value corresponding to the high-fidelity model Y t A high-fidelity verification sample dataset was obtained. xt , Y t ].
[0050] Step S3: Based on the low-fidelity training sample dataset [ x l , Y l Use the Kriging model to build a low-fidelity model.
[0051] Step S4: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ], establish the predicted mean of the hyperplane model y line The specific expression.
[0052] Step S5: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ],as well as y line and y l Develop a multifidelity model to obtain a specific expression for the predicted mean.
[0053] Step S6: Solve for the hyperparameters of the established multifidelity model using maximum likelihood estimation.
[0054] Step S7: Based on the process variance calculation results obtained from the model hyperparameters, further obtain the estimated value of the predicted mean and the predicted root mean square error of the multi-fidelity model, and output the final multi-fidelity surrogate model based on hyperplane correction. Finally, perform accuracy prediction on the high-fidelity validation samples to obtain the error evaluation results of maximum relative error (MAE) and root mean square error (RMSE). The method of this invention is compared with the Co-Kriging model, Co-RBF model, HK model, and IHK model. All methods use the same high-fidelity training sample dataset, low-fidelity training sample dataset, and high-fidelity validation sample dataset. The obtained high-fidelity model prediction results are as follows: Figure 3As shown in the figure, the RMSE and MAE results of the method proposed in this invention both achieve the minimum values, indicating that the prediction accuracy of this invention is superior to the other four methods. Simultaneously, the box plot of the method proposed in this invention achieves smaller widths and values under both error indices, indicating that the method has good robustness and minimal error. Therefore, the multi-fidelity model construction method based on hyperplane correction proposed in this invention can obtain prediction results with higher robustness and accuracy.
[0055] 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. A method for constructing a multi-fidelity model based on hyperplane correction, characterized in that, The multi-fidelity model construction method includes the following steps: Step S1: Obtain the corresponding high-fidelity model in the simulation software. g e ( x Low-fidelity model g l ( x And the sample space of the input variables; respectively obtain high-fidelity input samples. x e Low-fidelity input samples x l High-fidelity input verification samples x t ; Step S2: Process high-fidelity input samples x e Obtain the output response value corresponding to the high-fidelity model Y e For low-fidelity input samples x l Obtain the output response value corresponding to the low-fidelity model Y l ; and thus obtain a high-fidelity training sample data set [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ]; Validation samples for high-fidelity input x t Obtain the output response value corresponding to the high-fidelity model Y t A high-fidelity verification sample dataset was obtained. x t , Y t ]; Step S3: Based on the low-fidelity training sample dataset [ x l , Y l Use the Kriging model to build a low-fidelity model; Step S4: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ], establish the predicted mean of the hyperplane model y line The specific expression; Step S5: Based on the high-fidelity training sample dataset [ x e , Y e ] and low-fidelity training sample data set[ x l , Y l ],as well as y line and y l Develop a multi-fidelity model and determine the specific expression for its predicted mean; Step S6: Solve for the hyperparameters of the established multifidelity model using maximum likelihood estimation; Step S7: Obtain the process variance calculation results based on the model hyperparameters, obtain the estimated value of the prediction mean and the prediction mean square error of the multifidelity model, and output the final multifidelity surrogate model based on hyperplane correction.
2. The method for constructing a multi-fidelity model based on hyperplane correction according to claim 1, characterized in that, In step S1, high-fidelity input samples are obtained using Latin hypercube experimental design. x e and low-fidelity input samples x l In addition, high-fidelity input validation samples were obtained using Latin hypercube experimental design. x t .
3. The method for constructing a multi-fidelity model based on hyperplane correction according to claim 2, characterized in that, Step S3 specifically involves: Step S3-1: According to Kriging theory, the low-fidelity model can be represented as the sum of a polynomial and a random distribution function: （1）； In the formula, This represents a low-fidelity model. x Indicates the input prediction sample; Represents the regression parameters of the low-fidelity model; This represents the regression term of the low-fidelity model; Let Kriging represent a random distribution function, i.e., the prediction uncertainty introduced by the Kriging model, whose distribution is: ;in, Indicates a normal distribution. This represents the mean squared error of the Kriging model's predictions; Step S3-2: The obtained low-fidelity model follows a normal distribution, and its prediction mean is... y l .
4. The method for constructing a multi-fidelity model based on hyperplane correction according to claim 3, characterized in that, In step S4, the predicted mean of the hyperplane model y line The specific expression is: （2）； In the formula, e Represents a unit vector; y l Indicates the predicted sample x The corresponding low-fidelity model prediction mean; y line This represents the mean of the hyperplane model predictions for the predicted sample.
5. The method for constructing a multi-fidelity model based on hyperplane correction according to claim 4, characterized in that, In step S4, the specific expression for predicting the mean is defined as follows: （3）； In the formula, This represents the predicted mean of the hyperplane model; This represents the regression parameters corresponding to the hyperplane model; This represents the error term in a multifidelity model. Step S5-1: The error term of the multifidelity model is represented by a stochastic process with the following distribution: （4）； In the formula, This represents the mean square error of prediction in a multifidelity model; , and Perform splicing, define Represented as , This represents a concatenated sample matrix; Step S5-2: Integrate all regression terms They are defined as follows: （5）； In the formula, w This represents the predicted mean of the hyperplane model; v This represents the regression parameters corresponding to the hyperplane model; Step S5-3: For A high-fidelity training sample, regression term matrix It is expressed as follows: （6）； In the formula, Indicates the first One training sample; This represents the mean of the hyperplane model predictions for the first training sample; This represents the mean of the hyperplane model predictions for the second training sample; Indicates the first The mean of the hyperplane model predictions for each training sample; and W The corresponding training sample error term is represented as: ,in This represents the error term of the hyperplane model for the first training sample; This represents the hyperplane model error term for the second training sample; Indicates the first Error term of the hyperplane model for each training sample; Step S5-4: The covariance function of the correlation coefficient of the error terms for different training samples is expressed as follows: （7）； In the formula, Indicates process variance; Hyperparameters Related functions; Represents the covariance function; a This represents the first training sample in the correlation function; b This represents the second training sample in the correlation function; express a Error term of the hyperplane model for training samples; express b Error term of the hyperplane model for training samples; Represents the expected function; specifically, it is derived from the first... i A spliced sample matrix of high-fidelity training samples and the j A spliced sample matrix of high-fidelity training samples The mathematical expression for the correlation function of the sample composition is: ,in n Indicates the sample dimension; k Indicates the dimension number; Indicates the first k Hyperparameters corresponding to each sample dimension; Indicates the first i The first concatenated sample matrix in the nth concatenated sample matrix k One dimension; Indicates the first j The first concatenated sample matrix in the nth concatenated sample matrix k One dimension; Step S5-5: Predict the sample x Correlation function between and high-fidelity training samples Represented as: （8）； In the formula, This represents the concatenated sample matrix of the first high-fidelity training sample; This represents the concatenated sample matrix of the second high-fidelity training sample; Indicates the first A spliced sample matrix of high-fidelity training samples; Step S5-6: Treat the predicted values of the predicted samples as a linear combination of the observed samples. The estimated mean value of the multifidelity model prediction is expressed as: （9）； In the formula, This represents the mean estimate of the multifidelity model prediction; This represents the output response value corresponding to the high-fidelity model. The corresponding weighting factor matrix; This represents the first weighting factor; This represents the second weighting factor; Indicates the first One weighting factor; Step S5-7: The prediction error of the multi-fidelity model is expressed as: （10）； In the formula, This represents the output response value corresponding to the high-fidelity model. The corresponding weighting factor matrix; This represents the training sample error term of the multifidelity model; This represents the error term in the prediction samples of the multifidelity model; Step S5-8: Under the condition of unbiased estimation, let Mean square error of multifidelity model Represented as: （11）； Step S5-9: In order to calculate c The Lagrange function can be constructed as follows: （12）； In the formula, λ Represents the Lagrange multipliers; Step S5-10: Solve the Lagrange linear equations. The final parameter results are as follows: （13）； Step S5-11: The estimated value of the predicted mean of the established multi-fidelity model is: （14）； In the formula, This represents an estimate of the predicted mean of the multifidelity model. , ; Mean squared error of prediction in multifidelity models The expression is: （15）； In the formula, Process variance Solve using maximum likelihood estimation.
6. The method for constructing a multi-fidelity model based on hyperplane correction according to claim 5, characterized in that, Step S6 specifically involves: Step S6-1: Define the joint probability density function of all high-fidelity training samples as follows: （16）； In the formula, This represents the joint probability density function of high-fidelity training samples. Indicates hyperparameters; Step S6-2: Treat the joint probability density function as a likelihood function and perform a maximization operation, respectively for... and Taking the partial derivatives, we can solve for: （17）； Substituting the result into the likelihood function, while ignoring the constant term, we get: （18）； The optimal values of hyperparameters are calculated using a pattern search algorithm.
7. The method for constructing a multi-fidelity model based on hyperplane correction according to claim 6, characterized in that, In step S7, the accuracy of the high-fidelity verification sample is predicted to obtain the error evaluation results of the maximum relative error (MAE) and root mean square error (RMSE).