Modeling method of point-by-point weighted mixed agent model based on mixed measure

Abstract

The invention discloses a modeling method of a point-by-point weighted mixed proxy model based on mixed measurement. The modeling method comprises the following steps: extracting a training sample; four sub-agent models are constructed, and hyper-parameter optimization is carried out; determining a local reference model and a global reference model of each sample point according to the LOO cross validation error and the GMSE error, and determining a global weight coefficient; determining the sample density; estimating local precision; determining a local weight coefficient; obtaining a local error measure-based preliminary mixed agent model; and obtaining a point-by-point weighted mixed proxy model based on the mixed measure. According to the method provided by the invention, each prediction point should be predicted by the corresponding local sub-agent model with the optimal prediction capability as far as possible, the local fitting precision is further improved, and meanwhile, through remixing with the global reference model with the optimal global fitting precision, the local fitting precision is further improved. And the global fitting precision of the pointwise weighted mixed agent model is further improved.

Description

Technical Field

[0001] The invention relates to the construction of an approximate relationship between input and output of a mechanical structure, and in particular to a modeling method of a point-by-point weighted hybrid proxy model based on a hybrid measure. Background Art

[0002] With the advancement of mathematics and computing technology, computer simulation technology has shown significant advantages in the field of mechanical design, and has gradually replaced traditional experimental methods, providing strong technical support for mechanical design optimization. However, computer simulation technology is not perfect, and its computational cost is often high, especially when dealing with complex high-fidelity engineering simulation models. The computational cost will increase dramatically and the design cycle will be lengthened, making it difficult to rely entirely on computer simulation for design optimization.

[0003] In order to solve the problem of high computational cost of computational simulation experiments, a mechanical design method that uses proxy models to replace expensive computational simulation models has emerged. This method not only effectively reduces the computational cost, but also shortens the design cycle to a great extent. Compared with traditional computational simulation models, proxy models exist in a purely mathematical form and can quickly and accurately predict and simulate results, providing more efficient and reliable technical support for mechanical design optimization. However, due to the complexity of actual engineering problems, how to select a suitable proxy model before design optimization is still a difficult problem in the application field of proxy models.

[0004] The hybrid surrogate model forms a new surrogate model by carefully screening and combining multiple surrogate model methods. A large number of studies have shown that using a hybrid surrogate model instead of a single surrogate model for modeling can effectively avoid the risk of the surrogate model method selection strategy caused by the differences in the factor surrogate model methods. This not only significantly improves the overall modeling accuracy of the model, but also enhances the robustness of the model. However, the performance of the hybrid surrogate model is not always better than the optimal sub-surrogate model of its composition. In addition, most hybrid models are global average hybrid surrogate models, and their weight coefficients are set as constants, which means that once the weight coefficients are determined, they remain unchanged throughout the design interval. This method of globally fixed weight coefficients cannot reveal the prediction accuracy of each component surrogate model in the combination in the local area, so it may not provide accurate predictions in some local areas. That is, the global average hybrid surrogate model mostly only focuses on the global error of the sample set to ensure the global accuracy of the surrogate model. This has certain requirements on the distribution of the selected sample points. When the sample points are sparse or some sample points are far away from the main part, the overall prediction accuracy will be greatly affected due to the neglect of local errors.

[0005] In order to improve the local prediction accuracy of the hybrid proxy model, researchers proposed a hybrid proxy model based on the weight coefficient changing with the local accuracy, called the point-by-point weighted hybrid proxy model. This adaptive intelligent hybrid proxy model method can flexibly adjust the weight coefficient according to the needs of different regions, thereby improving the accuracy of prediction. However, they may have poor global fitting accuracy. At the same time, most of the current adaptive weight coefficients require optimization methods to assist in searching, especially for high-dimensional problems, and their computational cost is relatively high.

[0006] Therefore, for complex engineering application problems, finding a modeling method for a point-by-point weighted hybrid proxy model that is applicable to both low-dimensional and high-dimensional problems, and comprehensively considering global and local errors to ensure that its weight coefficients can ensure that the model has sufficient global and local accuracy, as well as low computational costs, is an important technical problem that needs to be solved in this field. Summary of the invention

[0007] The purpose of the present invention is to provide a modeling method of a point-by-point weighted hybrid proxy model based on hybrid measurement to address the problems of low accuracy, difficult calculation of weight coefficients and high cost of the existing hybrid proxy model.

[0008] The technical solution adopted by the present invention to achieve the above-mentioned purpose is:

[0009] A modeling method of a point-by-point weighted hybrid proxy model based on a hybrid measure comprises the following steps:

[0010] Step 1: Use Latin hypercube sampling to extract training samples in the variable space, and obtain the true response value through simulation analysis;

[0011] Step 2: Using the initial sample points and the corresponding true values, four sub-proxy models, PRS, RBF, KRG, and SVR, are constructed and hyper-parameter optimization is performed;

[0012] Step 3: According to the LOO cross-validation error and GMSE error, determine the local benchmark model and the global benchmark model for each sample point, and determine the global weight coefficient;

[0013] Step 4: Determine sample density based on crowding distance;

[0014] Step 5: Determine the local accuracy coefficient based on the local error measure;

[0015] Step 6: Determine the local weight coefficient of each sub-agent model;

[0016] Step 7: Determine a preliminary hybrid proxy model based on the local error measure;

[0017] Step 8: Determine the point-wise weighted hybrid surrogate model based on the hybrid measure.

[0018] Described step 1 specifically comprises:

[0019] The Latin hypercube sampling method is used to extract training samples in the original variable space and form the initial training point set x (j) (j=1,…,N); N represents the number of training samples;

[0020] The training samples are simulated and analyzed to obtain the true response value g(x (j) ), forming a database DB[x (j) |g(x (j) )](j=1,…,N); where x=(x1,…,x D ) is the D-dimensional input variable.

[0021] Step 2 uses the initial sample points and the corresponding true values ​​to construct the PRS sub-agent model, which specifically includes:

[0022] Step 2.1.1: Determine the data dimension D, select the basis function as a complete quadratic polynomial, perform multivariate polynomial regression analysis on the training sample points in the database DB and their true response values ​​according to the least squares method, and determine the regression function;

[0023] Step 2.1.2: Perform variance and error statistical analysis on the regression function to determine the PRS model, whose basic form is:

[0024]

[0025] Among them, y1(x) is the prediction function of the objective function in the PRS model, x i (i=1,…,D) is the i-th component of the D-dimensional variable, and β is the unknown weight coefficient of the prediction function;

[0026] Step 2 uses the initial sample points and the corresponding true values ​​to build the RBF sub-agent model and perform hyperparameter optimization, specifically including:

[0027] Step 2.2.1: Determine the data dimension D and select the kernel function type as Gaussian function, which is as follows:

[0028]

[0029] Among them, r is the distance norm between the location point and the sample point, and the Euclidean distance || xx is used. i ||, is a commonly used distance kernel function, σ is a partial expansion constant, σ>0, which is used to control the shape of the kernel function.

[0030] Step 2.2.2: Use radial function to linearly weight the sample points and determine the RBF model, whose basic form is:

[0031]

[0032] Among them, y2(x) is the prediction function of the objective function in the RBF model, x i (i=1,…,D) is the i-th component of the D-dimensional variable, is the distance kernel function, m is the number of kernel functions, λ is the linear combination weight coefficient; r is the distance norm between the position point and the center point (sample), and the Euclidean distance || xx is used. i ||;

[0033] Step 2 uses the initial sample points and the corresponding true values ​​to construct the KRG sub-agent model, which specifically includes:

[0034] Step 2.3.1: Determine the data dimension D, select the regression function of the KRG model as the quadratic polynomial model, and the correlation model function as the most typical Gaussian function;

[0035] Step 2.3.2: Define the value of theta and the thresholds lob and upb, where theta represents the hyperparameter of the selected correlation function;

[0036] Step 2.3.3: Perform KRG model training on the training sample points and their true response values ​​in the database DB. The basic form is:

[0037]

[0038] Among them, y3(x) is the prediction function of the objective function in the KRG model, β i is the regression coefficient, f i (x) is the regression function, Z l (x) has a mean of 0 and a variance of σ 2 A stationary random process;

[0039] In step 2, using the initial sample points and the corresponding true values, constructing the SVR sub-agent model and performing hyperparameter optimization specifically include:

[0040] Step 2.4.1: Determine the data dimension D, select e-SVR, and the kernel function type is RBF radial basis function;

[0041] Step 2.4.2: Define the loss function c and the value of the gamma function g in the kernel function, where c and g represent the selected hyperparameters;

[0042] Step 2.4.3: Perform SVR model training on the training sample points and their true response values ​​in the database DB. The basic form is:

[0043]

[0044] Among them, y4(x) is the prediction function of the objective function in the SVR model, k(x i T x) = θ(x i T )θ(x j ) is the kernel function.

[0045] Step 2.4.4: Initialize the particle swarm. The position of each particle in the particle swarm is the value (hyperparameter) of the loss function c and the gamma function g in the kernel function, and the speed is the change direction and rate of the hyperparameter. Determine the number of particles in the particle swarm and the initial position and initial speed of each particle. The optimization goal is to minimize the cross-validation error of the SVR sub-agent model and set the convergence conditions in the optimization process. After completing the particle swarm optimization, the optimal values ​​of the loss function c and the gamma function g in the kernel function are obtained, and the SVR sub-agent model is rebuilt.

[0046] The local benchmark model of each sample point is determined according to the LOO cross-validation error in step 3, specifically including:

[0047] Step 3.1.1: For any training sample point x (j) (j=1,…,N), four sub-agent models are constructed for LOO cross-validation respectively, and their LOO cross-validation error is obtained. The expression is as follows:

[0048]

[0049] in represents the LOO cross-validation error of the k-th sub-agent model at the j-th sample point, y (j) is the true response value of the model at this sample point, is the predicted response value of the kth sub-agent model at this sample point.

[0050] Step 3.1.2: For each sample point x (j) (j=1,…,N), select the sub-proxy model with the smallest LOO cross-validation error as the local benchmark model μ(x (i) );

[0051] Step 3 determines the global benchmark model and the global weight coefficient according to the LOO generalized mean square cross validation error, specifically including:

[0052] Step 3.2.1: Perform LOO cross validation again for the four sub-agent models and calculate the GMSE error. The expression is as follows:

[0053] GMSE k represents the GMSE error of the k-th sub-agent model, y (j) is the true response value of the model at the jth sample point, is the predicted response value of the kth sub-agent model at the sample point, and N represents the number of training points.

[0054] Step 3.2.2: Sort all sub-proxy models from low to high according to the GMSE error results. The sub-proxy model with the smallest GMSE error and the highest global accuracy is selected as the global benchmark model f g (x), and the global weight coefficient is calculated at the same time, and its expression is as follows:

[0055]

[0056] Among them, w g is the global weight coefficient, where GMSE1 is the smallest GMSE value and GMSE2 is the second smallest GMSE value. Step 4 determines the sample density based on the crowding distance, specifically including:

[0057] Step 4.1: For any prediction point x (i) , calculate its correlation with all training sample points x (j) The sum of the Euclidean distances D i , defined as the crowding distance, is expressed as follows:

[0058]

[0059] Where D i represents the crowding distance of the i-th prediction point, using the Euclidean distance, x (j) represents the jth training point, x (i) represents the i-th prediction point, and N is the number of training points.

[0060] Step 4.2: Calculate the sample density ρ at any prediction point i , which is expressed as follows:

[0061]

[0062] Step 4.3: Normalize the sample density to obtain the normalized sample density Nρ at any prediction point i , which is expressed as follows:

[0063]

[0064] The step 5 of determining the local accuracy coefficient based on the local error measure specifically includes:

[0065] For any prediction point x (i) , calculate the local accuracy coefficient of each sub-agent model at this point, and its expression is as follows:

[0066] Where P k (x (i) ) represents the local accuracy coefficient of the kth sub-agent model at the i-th prediction point, y k (x (i) ) represents the predicted response value of the kth sub-agent model at the i-th prediction point, μ(x (i) ) represents the predicted response value of the local benchmark model at the i-th prediction point, and the sensitivity parameter σ 2 The calculation formula is as follows:

[0067]

[0068] Where σ1(x (i) ) represents the sensitivity parameter at the i-th prediction point, Nρ i is the normalized sample density at the i-th prediction point.

[0069] Step 6 determines the local weight coefficient of each sub-agent model, specifically including:

[0070] For any prediction point x (i) , by normalizing the local precision coefficient of each sub-proxy model in the sub-model library, the local weight coefficient corresponding to each sub-proxy model can be calculated. The expression is as follows:

[0071]

[0072] where w localk (x (i) ) represents the local weight coefficient of the sub-agent model k at the i-th prediction point, P k (x (i) ) represents the local accuracy coefficient of the sub-agent model k at the i-th prediction point, n s Indicates the number of sub-agent models in the model library.

[0073] Determining the preliminary hybrid proxy model based on the local error measure in step 7 specifically includes:

[0074] Through each sub-agent model and the local weight coefficient, the preliminary hybrid agent model based on the local error measure is determined, and its expression is as follows:

[0075]

[0076] where f PHSM represents the preliminary mixed agent model, y kRepresents the sub-agent model k, w local represents the local weight coefficient of the sub-agent model k, n s Indicates the number of sub-agent models in the model library.

[0077] Determining the point-by-point weighted hybrid proxy model based on the hybrid measure in step 8 specifically includes:

[0078] According to the preliminary hybrid surrogate model based on the local error measure f PHSM , global benchmark model f g And the global weight coefficient w g , determine the point-wise weighted hybrid surrogate model f based on the hybrid measure PWHSMHM , which is expressed as follows:

[0079] f PWHSMHM =w g f G +(1-w g )f PHSM

[0080] where f PWHSMHM represents the point-wise weighted hybrid surrogate model based on the hybrid measure, f g represents the global benchmark model, f PHSM Represents a preliminary hybrid agent model.

[0081] Compared with the prior art, the present invention has the following beneficial effects:

[0082] (1) The present invention uses the LOO cross-validation error as the basis for selecting the local benchmark model, and calculates the local weight coefficient of each sub-proxy model based on the local benchmark model of each sample point, ensuring that each prediction point should be predicted as much as possible by its corresponding local sub-proxy model with the best prediction ability, thereby improving the local prediction accuracy of the point-by-point weighted hybrid proxy model, without the need for optimization calculation, and with low time cost.

[0083] (2) The present invention uses GMSE error as the basis for selecting the global benchmark model and calculating the global weight coefficient, thereby ensuring the global prediction accuracy of the point-by-point weighted hybrid proxy model.

[0084] (3) The hybrid measure of the present invention integrates the global error measure and the local error measure, thereby ensuring the overall prediction accuracy of the point-by-point weighted hybrid proxy model.

[0085] In summary, the present invention has the advantages of simple logic and effective improvement of the overall prediction accuracy of the hybrid agent model, and has high practical value and promotion value in the technical field of hybrid agent model construction. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1It is a schematic diagram of the process of the present invention.

[0087] Figure 2 The invention is based on the point-by-point weighted hybrid proxy model and four sub-proxy models in a specific application example to model 40 standard test functions. 2 Box plot of .

[0088] Figure 3 The invention is based on the point-by-point weighted hybrid proxy model and four sub-proxy models in a specific application example to model 40 standard test functions. 2 Mean and R 2 >0.8 quantity comparison chart.

[0089] Figure 4 It is a comparison chart of the NRMSE mean values ​​of the modeling performance indicators of 40 standard test functions of the point-by-point weighted hybrid proxy model based on hybrid measurement and four sub-proxy models in a specific application example of the present invention. DETAILED DESCRIPTION

[0090] The following is a clear and complete description of the technical solutions in the embodiments of the present invention in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0091] In this embodiment, the term "and / or" is merely a term used to describe the association relationship of associated objects, indicating that three relationships may exist. For example, A and / or B may represent three situations: A exists alone, A and B exist at the same time, and B exists alone.

[0092] The terms "first" and "second" in the description and claims of this embodiment are used to distinguish different objects rather than to describe a specific order of objects. For example, a first target object and a second target object are used to distinguish different target objects rather than to describe a specific order of target objects.

[0093] In the embodiments of the present application, words such as "exemplary" or "for example" are used to indicate examples, illustrations or descriptions. Any embodiment or design described as "exemplary" or "for example" in the embodiments of the present application should not be interpreted as being more preferred or more advantageous than other embodiments or designs. Specifically, the use of words such as "exemplary" or "for example" is intended to present related concepts in a specific way.

[0094] In the description of the embodiments of the present application, unless otherwise specified, the meaning of "multiple" refers to two or more than two. For example, multiple processing units refer to two or more processing units; multiple systems refer to two or more systems.

[0095] like Figure 1-4 As shown, the embodiment of the present invention proposes a modeling method of a point-by-point weighted hybrid proxy model based on a hybrid measure, such as Figure 1 As shown, in this embodiment, 40 standard test functions from low dimension (1 dimension) to high dimension (16 dimensions) are modeled, and the method of the present invention is used to determine the point-by-point weighted hybrid proxy model based on the hybrid measure and perform accuracy verification, so as to understand the specific modeling process and technical advantages of the technical method of the present disclosure, which specifically includes the following steps:

[0096] Step 1: Use Latin hypercube sampling to extract training samples in the original variable space and form an initial training point set x (j) (j=1,…,N); N represents the number of training samples, N=10D, D is the dimension of the input variable; the training samples are simulated and analyzed to obtain the true response value g(x (j) ), forming a database DB[x (j) |g(x (j) )](j=1…,,N);

[0097] Where x=(x1,…,x D ) is the D-dimensional input variable.

[0098] The 40 standard test functions are shown in Table 1:

[0099] Table 1 40 standard test functions

[0100]

[0101]

[0102]

[0103] Step 2: Using the initial sample points and the corresponding true values, the PRS sub-surrogate model is constructed using the fitnlm toolbox in MATLAB, including:

[0104] Step 2.1.1: Determine the data dimension D, select the basis function as a complete quadratic polynomial, perform multivariate polynomial regression analysis on the training sample points in the database DB and their true response values ​​according to the least squares method, and determine the regression function;

[0105] Step 2.1.2: Perform variance and error statistical analysis on the regression function to determine the PRS model, whose basic form is:

[0106]

[0107] Among them, y1(x) is the prediction function of the objective function in the PRS model, x i (i=1,…,D) is the i-th component of the D-dimensional variable, and β is the unknown weight coefficient of the prediction function;

[0108] Using the initial sample points and the corresponding true values, the RBF sub-agent model is constructed in MATLAB, including:

[0109] Step 2.2.1: Determine the data dimension D and select the kernel function type as Gaussian function, which is as follows:

[0110]

[0111] Among them, r is the distance norm between the location point and the sample point, and the Euclidean distance || xx is used. i ||, is a commonly used distance kernel function, σ is a partial expansion constant, σ>0, which is used to control the shape of the kernel function.

[0112] Step 2.2.2: Use radial function to linearly weight the sample points and determine the RBF model, whose basic form is:

[0113]

[0114] Among them, y2(x) is the prediction function of the objective function in the RBF model, x i (i=1,…,D) is the i-th component of the D-dimensional variable, is the distance kernel function, m is the number of kernel functions, λ is the linear combination weight coefficient; r is the distance norm between the position point and the center point (sample), and the Euclidean distance || xx is used. i ||;

[0115] Using the initial sample points and the corresponding true values, the KRG sub-agent model is constructed using the DACE toolbox in MATLAB, including:

[0116] Step 2.3.1: Determine the data dimension D, select the regression function of the KRG model as the quadratic polynomial model, and the correlation model function as the most typical Gaussian function;

[0117] Step 2.3.2: Define the value of theta and the thresholds lob and upb, where theta represents the hyperparameter of the selected correlation function;

[0118] Step 2.3.3: Perform KRG model training on the training sample points and their true response values ​​in the database DB. The basic form is:

[0119]

[0120] Among them, y3(x) is the prediction function of the objective function in the KRG model, β i is the regression coefficient, f i (x) is the regression function, Z l (x) has a mean of 0 and a variance of σ 2 A stationary random process;

[0121] Using the initial sample points and the corresponding true values, the SVR sub-agent model is constructed using the SVR support vector regression toolbox in MATLAB, and the particle swarm algorithm is used to optimize the hyperparameters, including:

[0122] Step 2.4.1: Determine the data dimension D, select e-SVR, and the kernel function type is RBF radial basis function;

[0123] Step 2.4.2: Define the loss function c and the value of the gamma function g in the kernel function, where c and g represent the selected hyperparameters;

[0124] Step 2.4.3: Perform SVR model training on the training sample points and their true response values ​​in the database DB. The basic form is:

[0125]

[0126] Among them, y4(x) is the prediction function of the objective function in the SVR model, k(x i T x) = θ(x i T )θ(x j ) is the kernel function.

[0127] Step 2.4.4: Initialize the particle swarm. The position of each particle in the particle swarm is the value (hyperparameter) of the loss function c and the gamma function g in the kernel function, and the speed is the change direction and rate of the hyperparameter. Determine the number of particles in the particle swarm and the initial position and initial speed of each particle. The optimization goal is to minimize the cross-validation error of the SVR sub-agent model and set the convergence conditions in the optimization process. After completing the particle swarm optimization, the optimal values ​​of the loss function c and the gamma function g in the kernel function are obtained, and the SVR sub-agent model is rebuilt.

[0128] In this embodiment, the number of particles in the particle swarm is set to 10, and the preset convergence condition is that the number of optimizations reaches a preset maximum number of iterations of 100 times.

[0129] Step 3: Determine the local benchmark model for each sample point based on the LOO cross-validation error, including:

[0130] Step 3.1.1: For any training sample point x (j) (j=1,…,N), four sub-agent models are constructed for LOO cross-validation respectively, and their LOO cross-validation error is obtained. The expression is as follows:

[0131]

[0132] in represents the LOO cross-validation error of the k-th sub-agent model at the j-th sample point, y (j) is the true response value of the model at this sample point, is the predicted response value of the kth sub-agent model at this sample point.

[0133] Step 3.1.2: For each sample point x (j) (j=1,…,N), select the sub-proxy model with the smallest LOO cross-validation error as the local benchmark model μ(x (i) );

[0134] The global benchmark model and global weight coefficient are determined according to the LOO generalized mean square cross validation error, specifically including:

[0135] Step 3.2.1: Perform LOO cross validation again for the four sub-agent models and calculate the GMSE error. The expression is as follows:

[0136] GMSE k represents the GMSE error of the k-th sub-agent model, y (j) is the true response value of the model at the jth sample point, is the predicted response value of the kth sub-agent model at the sample point, and N represents the number of training points.

[0137] Step 3.2.2: Sort all sub-proxy models from low to high according to the GMSE error results. The sub-proxy model with the smallest GMSE error and the highest global accuracy is selected as the global benchmark model f g (x), and the global weight coefficient is calculated at the same time, and its expression is as follows:

[0138]

[0139] Among them, w g is the global weight coefficient, where GMSE1 is the smallest GMSE value and GMSE2 is the second smallest GMSE value. Step 4: Determine the sample density based on the crowding distance, specifically including:

[0140] Step 4.1: For any prediction point x(i) , calculate its correlation with all training sample points x (j) The sum of the Euclidean distances D i , defined as the crowding distance, is expressed as follows:

[0141]

[0142] Where D i represents the crowding distance of the i-th prediction point, using the Euclidean distance, x (j) represents the jth training point, x (i) represents the i-th prediction point, and N is the number of training points.

[0143] Step 4.2: Calculate the sample density ρ at any prediction point i , which is expressed as follows:

[0144]

[0145] Step 4.3: Normalize the sample density to obtain the normalized sample density Nρ at any prediction point i , which is expressed as follows:

[0146]

[0147] Step 5: Obtain the local accuracy coefficient based on the local uncertainty, including:

[0148] For any prediction point x (i) , calculate the local accuracy coefficient of each sub-agent model at this point, and its expression is as follows:

[0149] Where P k (x (i) ) represents the local accuracy coefficient of the kth sub-agent model at the i-th prediction point, y k (x (i) ) represents the predicted response value of the kth sub-agent model at the i-th prediction point, μ(x (i) ) represents the predicted response value of the local benchmark model at the i-th prediction point, and the sensitivity parameter σ 2 The calculation formula is as follows:

[0150]

[0151] Where σ1(x (i) ) represents the sensitivity parameter at the i-th prediction point, Nρ i is the normalized sample density at the i-th prediction point.

[0152] Step 6: Determine the local weight coefficient of each sub-agent model, including:

[0153] For any prediction point x (i) , by normalizing the local precision coefficient of each sub-proxy model in the sub-model library, the local weight coefficient corresponding to each sub-proxy model can be calculated. The expression is as follows:

[0154]

[0155] where w localk (x (i) ) represents the local weight coefficient of the sub-agent model k at the i-th prediction point, P k (x (i) ) represents the local accuracy coefficient of the sub-agent model k at the i-th prediction point, n s Indicates the number of sub-agent models in the model library.

[0156] Step 7: Determine a preliminary hybrid proxy model based on local error measures, including:

[0157] Through each sub-agent model and the local weight coefficient, the preliminary hybrid agent model based on the local error measure is determined, and its expression is as follows:

[0158]

[0159] where f PHSM represents the preliminary mixed agent model, y k Represents the sub-agent model k, w local represents the local weight coefficient of the sub-agent model k, n s Indicates the number of sub-agent models in the model library.

[0160] Step 8: Determine a point-by-point weighted hybrid proxy model based on the hybrid measure, including:

[0161] According to the preliminary hybrid surrogate model based on the local error measure f PHSM , global benchmark model f g And the global weight coefficient w g , determine the point-wise weighted hybrid surrogate model f based on the hybrid measure PWHSMHM , which is expressed as follows:

[0162] f PWHSMHM =w g f g +(1-w g )f PHSM

[0163] where f PWHSMHM represents the point-wise weighted hybrid surrogate model based on the hybrid measure, f g represents the global benchmark model, f PHSMRepresents a preliminary hybrid agent model.

[0164] Step 9: Conduct accuracy verification of the proxy model, including:

[0165] The above 40 standard test functions were fitted using four sub-surrogate models and a point-wise weighted hybrid surrogate model. Then, 500D test points were randomly generated by Latin hypercube sampling, and the test points were predicted using the constructed surrogate model. The prediction performance was calculated by calculating the coefficient of determination (R 2 ) and normalized root mean square error (NRMSE) are evaluated, and the expressions are as follows:

[0166]

[0167] Among them, N test is the number of test set samples, y i is the true value of the i-th test point, is the predicted value of the i-th test point. Tables 2 to 5 and Figures 2 to 4 The test results of the pointwise weighted hybrid surrogate model (PWHSMHM) based on hybrid measure using the proposed method for 40 standard test functions are given, as well as a comparison with four sub-surrogate models.

[0168] R 2 The closer the value is to 1, the smaller the overall error between the proxy model and the true model, and the higher the approximation. Figures 2 to 4 It can be seen that for 40 standard test functions, the pointwise weighted hybrid surrogate model (PWHSMHM) based on hybrid measure has a higher degree of approximation to the true model than the four sub-surrogate models as a whole.

[0169] The smaller the NRMSE value, the higher the fitting accuracy of the model. Figure 4 It can be seen that for 40 standard test functions, the fitting accuracy of the pointwise weighted hybrid surrogate model (PWHSMHM) based on hybrid measure to the true model is generally higher than that of the four sub-surrogate models.

[0170] R 2 The test results are consistent with those of NRMSE, proving that the fitting accuracy and robustness of the point-by-point weighted hybrid surrogate model (PWHSMHM) based on hybrid measure are better than those of the classic sub-surrogate model method. It can be seen that the integrated error measure that comprehensively considers global accuracy and local accuracy is of great significance for improving the prediction accuracy of the hybrid surrogate model. It should be stated that some surrogate models, especially for high-dimensional problems, cannot achieve approximate fitting, R 2 The calculated value is negative, but this negative value only represents a failure of fitting and has no actual statistical significance. It will also affect the analysis of R 2The mean affects all R values ​​less than 0. 2 All are set to 0, indicating that the fitting failed.

[0171] Table 2 Test results: R 2 index

[0172]

[0173] Table 3R 2 Indicator statistics results

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[0175] Table 4 Test results: NRMSE index

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[0178] Table 5 NRMSE indicator statistical results

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[0180] Finally, it should be noted that the above embodiments are only used to illustrate and explain the present invention, and are not intended to limit the present invention to the scope of the described embodiments. In addition, those skilled in the art can understand that the present invention is not limited to the above embodiments, and more variations and modifications can be made according to the teachings of the present invention, and these variations and modifications all fall within the scope of the protection claimed by the present invention.

Claims

1. A modeling method of a point-wise weighted hybrid proxy model based on a hybrid measure, characterized in that: The steps include: Step 1: Use Latin hypercube sampling to extract training samples in the variable space, and obtain the true response value through simulation analysis; Step 2: Using the initial sample points and the corresponding true response values, construct the PRS, RBF, KRG, and SVR sub-proxy models and perform hyperparameter optimization; Step 3: According to the LOO cross-validation error and GMSE error, determine the local benchmark model and the global benchmark model for each sample point, and determine the global weight coefficient; Step 4: Determine sample density based on crowding distance; Step 5: Determine the local accuracy coefficient based on the local error measure; Step 6: Determine the local weight coefficient of each sub-agent model; Step 7: Determine a preliminary hybrid proxy model based on the local error measure; Step 8: Determine the point-wise weighted hybrid surrogate model based on the hybrid measure.

2. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 1 is characterized in that: The step 1 specifically includes: The Latin hypercube sampling method is used to extract training samples in the original variable space and form the initial training point set x (j) (j=1,…,N); N represents the number of training samples; The training samples are simulated and analyzed to obtain the true response value g(x (j) ), forming a database DB[x (j) |g(x (j) )](j=1,…,N); where x=(x1,…,x D ) is the D-dimensional input variable.

3. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 2 is characterized in that: The step 2 specifically includes: Construct the PRS sub-agent model: Step 2.1.1: Determine the data dimension D, select the basis function as a complete quadratic polynomial, perform multivariate polynomial regression analysis on the training sample points in the database DB and their true response values ​​according to the least squares method, and determine the regression function; Step 2.1.2: Perform variance and error statistical analysis on the regression function to determine the PRS model, whose basic form is: Among them, y1(x) is the prediction function of the objective function in the PRS model, x i (i=1,…,D) is the i-th component of the D-dimensional variable, and β is the unknown weight coefficient of the prediction function; Construct RBF sub-agent model: Step 2.2.1: Determine the data dimension D and select the kernel function type as Gaussian function, which is as follows: Among them, r is the distance norm between the location point and the sample point, and the Euclidean distance || xx is used. i ||, is a commonly used distance kernel function, σ is a partial expansion constant, σ>0, which is used to control the shape of the kernel function; Step 2.2.2: Use radial function to linearly weight the sample points and determine the RBF model, whose basic form is: Among them, y2(x) is the prediction function of the objective function in the RBF model, x i (i=1,…,D) is the i-th component of the D-dimensional variable, is the distance kernel function, m is the number of kernel functions, λ is the linear combination weight coefficient; r is the distance norm between the position point and the center point, and the Euclidean distance || xx is used. i ||; Construct KRG sub-agent model: Step 2.3.1: Determine the data dimension D, select the regression function of the KRG model as the quadratic polynomial model, and the correlation model function as the Gaussian function; Step 2.3.2: Define the value of theta and the thresholds lob and upb, where theta represents the hyperparameter of the selected correlation function; Step 2.3.3: Perform KRG model training on the training sample points and their true response values ​​in the database DB. The basic form is: Among them, y3(x) is the prediction function of the objective function in the KRG model, β i is the regression coefficient, f i (x) is the regression function, Z l (x) has a mean of 0 and a variance of σ 2 A stationary random process; Construct the SVR sub-agent model: Step 2.4.1: Determine the data dimension D, select e-SVR, and the kernel function type is RBF radial basis function; Step 2.4.2: Define the loss function c and the value of the gamma function g in the kernel function, where c and g represent the selected hyperparameters; Step 2.4.3: Perform SVR model training on the training sample points and their true response values ​​in the database DB. The basic form is: Among them, y4(x) is the prediction function of the objective function in the SVR model, k(x i T x) = θ(x i T )θ(x j ) is the kernel function; Step 2.4.4: Initialize the particle swarm. The position of each particle in the particle swarm is the value of the loss function c and the gamma function g in the kernel function. c and g represent the selected hyperparameters. The speed is the direction and rate of change of the hyperparameters. Determine the number of particles in the particle swarm and the initial position and initial speed of each particle. The optimization goal is to minimize the cross-validation error of the SVR sub-agent model and set the convergence conditions in the optimization process. After completing the particle swarm optimization, obtain the optimal values ​​of the loss function c and the gamma function g in the kernel function, and rebuild the SVR sub-agent model.

4. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 3 is characterized in that: The step 3 specifically includes: Step 3.1.1: For any training sample point x (j) (j=1,…,N), four sub-agent models are constructed for LOO cross-validation respectively, and their LOO cross-validation error is obtained. The expression is as follows: in represents the LOO cross-validation error of the k-th sub-agent model at the j-th sample point, y (j) is the true response value of the model at this sample point, is the predicted response value of the kth sub-agent model at the sample point; Step 3.1.2: For each sample point x (j) (j=1,…,N), select the sub-proxy model with the smallest LOO cross-validation error as the local benchmark model μ(x (i) ); Step 3.2.1: Perform LOO cross validation again for the four sub-agent models and calculate the GMSE error. The expression is as follows: GMSE k represents the GMSE error of the k-th sub-agent model, y (j) is the true response value of the model at the jth sample point, is the predicted response value of the kth sub-agent model at the sample point, and N represents the number of training points; Step 3.2.2: Sort all sub-proxy models from low to high according to the GMSE error results. The sub-proxy model with the smallest GMSE error and the highest global accuracy is selected as the global benchmark model f g (x), and the global weight coefficient is calculated at the same time, and its expression is as follows: Among them, w g is the global weight coefficient, where GMSE1 is the smallest GMSE value and GMSE2 is the second smallest GMSE value.

5. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 4 is characterized in that: The step 4 specifically includes: Step 4.1: For any prediction point x (i) , calculate its correlation with all training sample points x (j) The sum of the Euclidean distances D i , defined as the crowding distance, is expressed as follows: Where D i represents the crowding distance of the i-th prediction point, using the Euclidean distance, x (j) represents the jth training point, x (i) represents the i-th prediction point, and N is the number of training points; Step 4.2: Calculate the sample density ρ at any prediction point i , which is expressed as follows: Step 4.3: Normalize the sample density to obtain the normalized sample density Nρ at any prediction point i , which is expressed as follows:

6. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 5 is characterized in that: The step 5 specifically includes: For any prediction point x (i) , calculate the local accuracy coefficient of each sub-agent model at this point, and its expression is as follows: Where P k (x (i) ) represents the local accuracy coefficient of the kth sub-agent model at the i-th prediction point, y k (x (i) ) represents the predicted response value of the kth sub-agent model at the i-th prediction point, μ(x (i) ) represents the predicted response value of the local benchmark model at the i-th prediction point, and the sensitivity parameter σ 2 The calculation formula is as follows: Where σ1(x (i) ) represents the sensitivity parameter at the i-th prediction point, Nρ i is the normalized sample density at the i-th prediction point.

7. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 6 is characterized in that: The step 6 specifically includes: For any prediction point x (i) , by normalizing the local precision coefficient of each sub-proxy model in the sub-model library, the local weight coefficient corresponding to each sub-proxy model is calculated. The expression is as follows: where w localk (x (i) ) represents the local weight coefficient of the sub-agent model k at the i-th prediction point, P k (x (i) ) represents the local accuracy coefficient of the sub-agent model k at the i-th prediction point, n s Indicates the number of sub-agent models in the model library.

8. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 7 is characterized in that: The step 7 specifically includes: Through each sub-agent model and the local weight coefficient, the preliminary hybrid agent model based on the local error measure is determined, and its expression is as follows: where f PHSM represents the preliminary mixed agent model, y k Represents the sub-agent model k, w local represents the local weight coefficient of the sub-agent model k, n s Indicates the number of sub-agent models in the model library.

9. The modeling method of the point-by-point weighted hybrid proxy model based on hybrid measure according to claim 8, characterized in that: The step 8 specifically includes: According to the preliminary hybrid surrogate model based on the local error measure f PHSM , global benchmark model f g And the global weight coefficient w g , determine the point-wise weighted hybrid surrogate model f based on the hybrid measure PWHSMHM , which is expressed as follows: f PWHSMHM =w g f g +(1-w g )f PHSM where f PWHSMHM represents the point-wise weighted hybrid surrogate model based on the hybrid measure, f g represents the global benchmark model, f PHSM Represents a preliminary hybrid agent model.

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