Adaptive combined proxy model modeling method based on fuzzy C-means clustering

Abstract

The invention discloses a self-adaptive combined agent model modeling method based on fuzzy C-means clustering, which is characterized in that a single agent model is constructed by distributing training points which most accord with the characteristics of each component model to each component model in a mode of carrying out block processing on original training point data by virtue of the idea of fuzzy C-means clustering; performing two times of model screening on a single agent model in an agent model library to ensure the matching between block data and a component model, finally selecting the Euclidean distance between the component model and a base model as the basis of weight coefficient distribution, and outputting a final model. According to the method, the precision of each component model during construction of the model library is improved, a complex training point set can be better dealt with, the calculation amount is reduced, and meanwhile, the calculated self-adaptive weight coefficient of each component model is more reasonable according to the regional distribution primary function. Therefore, while the modeling efficiency is effectively improved, the overall precision and the local precision of the model can be ensured, and the calculation cost is reduced.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of proxy model, and particularly relates to a self-adaptive combined proxy model modeling method based on fuzzy C-means clustering. BACKGROUND

[0002] A proxy model is a fast substitute model of a real model. In practical applications, the proxy model can be used to replace complex numerical simulation calculation and expensive physical analysis between input and output, to perform subsequent optimization design and analysis, to solve the problem of dimension disaster to a certain extent, to improve the solving efficiency of engineering problems, and to shorten the product life cycle. The combined proxy model gradually replaces the single proxy model in engineering problems in recent years due to its significant advantages in prediction performance, robustness, generalization ability and the like. The combined proxy model superimposes multiple single proxy models with specific weight coefficients. The core is to determine appropriate weight coefficients. Compared with the combined proxy model with fixed weight, the self-adaptive weight proxy model has application advantages in engineering due to its dynamic adjustment characteristics. The self-adaptive weight model determines the local accuracy of the proxy model in the confidence domain based on the crowding distance, and allocates the weight accordingly. Subsequent scholars have made a series of improvements around it. For example, the two-stage self-adaptive combined proxy model proposed in recent years calculates the cross-validation error in the entire design domain to screen the base model, and simultaneously performs self-adaptive adjustment according to the characteristics of different regions to ensure local optimization in local prediction. However, the two-stage self-adaptive combined proxy model has high calculation cost, especially in the case of high-dimensional and high nonlinearity problems, the modeling efficiency is greatly reduced. At the same time, due to the characteristics that the base model remains unchanged in the entire design domain, the accuracy of different regions also fluctuates greatly. The reason is that the calculation of the cross-validation error is time-consuming, and the screening criteria of the base model ignore the characteristics of the single component model, which is difficult to fully play the performance in the face of complex training points. Therefore, in order to better play the characteristics of the single model in the combined proxy model and improve the modeling efficiency of the combined proxy model, it is necessary to propose a more reliable combined proxy model modeling method to make up for the defects of the two-stage self-adaptive combined proxy model. SUMMARY

[0003] In view of the problems existing in the existing self-adaptive combined proxy model, the application provides a self-adaptive combined proxy model modeling method based on fuzzy C-means clustering. The idea of fuzzy C-means clustering is borrowed. The original training point data is processed by block processing. The training points most suitable for the characteristics of each component model are allocated to construct a single proxy model. Specifically, the matching of the block data and the component model is ensured by performing two model screenings on the single proxy models in the proxy model library. Finally, the Euclidean distance between the component model and the base model (base function) is selected as the basis for weight coefficient distribution, and the final model is output.

[0004] The technical scheme adopted by the present application is as follows:

[0005] The present application provides an adaptive combined surrogate model modeling method based on fuzzy C-means clustering, comprising the following steps:

[0006] Step one, a set of training points is generated by Latin hypercube sampling, the design variable true response corresponding to the training points is obtained by simulation, and an initial training set including the training points and the true response is obtained. The membership degree C of each training point to the cluster center is obtained by using fuzzy C-means clustering, and the initial training set is divided into O sub-training sets.

[0007] Step two, a surrogate model library containing multiple single surrogate models is constructed.

[0008] Step three, each sub-training set after blocking is used to train each single surrogate model in the surrogate model library, and each sub-training set obtains a sub-model library after training. O sub-model libraries are obtained by traversing the O sub-training sets. Then, the global error of each single surrogate model in the sub-model library is calculated according to the prediction sum criterion, and the single surrogate model with the smallest global error in the sub-model library is selected as the component model of the sub-model library.

[0009] Step four, the cross-validation error of each component model is calculated, a cross-validation error threshold is set, the component models with cross-validation error higher than the threshold are removed, and a new model library is constructed.

[0010] Step five, a clustering method is used to process all basis functions, the design domain is divided into O block regions, the corresponding basis functions of each block region are obtained, the adaptive weight coefficients of each component model are determined according to the Euclidean distance between the component models in the new model library and the basis functions, and the final combined surrogate model is output.

[0011] Preferably, the surrogate model library contains surrogate models constructed by polynomial response surface method, radial basis function method, Kriging method and support vector regression method.

[0012] More preferably, the radial basis function method is divided into multiple quadratic radial basis function method and thin plate spline radial basis function method.

[0013] Preferably, the formula for calculating the global error of the single surrogate model in the sub-model library is:

[0014]

[0015] Wherein, n is the number of training points in the sub-training set; y i is the true response of the i th training point in the sub-training set; is the predicted response of the i th training point in the sub-training set by the single surrogate model trained by the set after deleting the i th training point.

[0016] More preferably, the cross-validation error of the component model is calculated as follows:

[0017]

[0018] Wherein: is the predicted response of the component model to the i-th training point in the sub-training set.

[0019] Preferably, the final combined proxy model:

[0020]

[0021] Wherein, is the predicted response of the final combined proxy model;

[0022] The weight coefficient of the i-th component model in the j-th block region:

[0023]

[0024] The kernel probability density function of the i-th component model in the new model library at the test point x in the j-th block region is calculated as follows:

[0025]

[0026] Wherein: is the predicted response of the i-th component model at the test point x in the j-th block region, y best_j (x) is the response of the basis function of the j-th block region at the test point x, K is the number of component models in the new model library, K≤O;

[0027] The local uncertainty s 2 (x) of each component model in the new model library is calculated as follows:

[0028]

[0029] Wherein: is the predicted standard deviation in the Gaussian process, R is the covariance matrix between the training points, and r is the covariance matrix between the training points and the test points.

[0030] The present application has the beneficial effects:

[0031] The application provides an adaptive combined proxy model modeling method based on fuzzy C-means clustering, which performs block processing on original training point data, and allocates appropriate training points to each component model to construct a single proxy model, thereby effectively improving the accuracy of each component model when constructing a model library, and better coping with complex training point sets. On the other hand, the step of using all training points to construct a proxy model before allocating weights for each component model is avoided, thereby reducing the amount of calculation. Meanwhile, the application divides the design domain into regions with the same number of original training point data, and allocates basis functions according to the regions, so that the adaptive weight coefficients of each component model calculated are more reasonable. The application can effectively improve the modeling efficiency, ensure the overall accuracy and local accuracy of the model, and reduce the calculation cost. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 The flowchart of the application.

[0033] Fig. 2(a), Fig. 2(b) and Fig. 2(c) are R 2 , NMAE and NRMSE comparison diagrams of the application and other combined models.

[0034] Fig. 3(a), Fig. 3(b) and Fig. 3(c) are R 2 , NMAE and NRMSE comparison diagrams of the application and other combined models under different sizes of initial training sets.

[0035] Fig. 4(a), Fig. 4(b) and Fig. 4(c) are R 2 , NMAE and NRMSE comparison diagrams of the application and other combined models under different dimensions. DETAILED DESCRIPTION

[0036] The application is further described below in combination with the drawings and specific implementation steps.

[0037] See Figure 1 , the adaptive combined proxy model modeling method based on fuzzy C-means clustering designed by the application mainly includes the following steps:

[0038] Step one: generating training points, processing the training points to obtain a sub-training set, specifically as follows:

[0039] (1) generating a set X t =[x1,x2,…,x s ] T, the real response Y of the design variables corresponding to the training points is obtained through simulation (such as finite element structure simulation, CFD fluid simulation, and of course simulation with real models) t = [y1, y2, …, y s ] T , and an initial training set is obtained.

[0040] (2) The membership C of each training point to the cluster center is obtained by using the FCM method (fuzzy C-means clustering), and the initial training set is divided into O sub-training sets (or component training sets, each sub-training set being a cluster). The cluster center and the membership C are optimized and iterated by optimizing the objective function. It should be noted that FCM allows a single training point to belong to multiple cluster centers, so the training points in each sub-training set may be repeated, which reflects the good adaptability of the FCM method to complex and fuzzy training point sets.

[0041] Step two: Establishing a proxy model library. This stage mainly includes the following steps: constructing a proxy model library containing multiple single proxy models, the proxy model library containing single proxy models suitable for high-dimensional, low-dimensional problems, linear and nonlinear problems. According to the single proxy models widely used at present, the most representative several single proxy models are selected, including the proxy models constructed by polynomial response surface method (PRS), radial basis function method (RBF), Kriging method (KRG) and support vector regression method (SVR). Among them, the radial basis function method is divided into multiple quadratic radial basis function method (RBF-MQ) and thin plate spline radial basis function method (RBF-TPS) due to the difference of its basis function.

[0042] Step three: First screening of the proxy model library. This stage mainly includes the following steps:

[0043] (1) Each single proxy model in the proxy model library is trained by using each sub-training set after blocking, and each sub-training set obtains a sub-model library after training. O sub-model libraries are obtained by traversing O sub-training sets.

[0044] (2) According to the prediction sum of squares (Prediction Sum of Squares, PRESS) criterion, the global error of each single proxy model in the sub-model library is calculated, and the specific formula is as follows:

[0045]

[0046] Where: n is the number of training points in the sub-training set; y i is the real response of the i-th training point in the sub-training set; The predicted response of the i-th training point in the sub-training set by the single surrogate model trained with the set after the i-th training point is deleted.

[0047] (3) The single surrogate model with the minimum global error in the sub-model library is selected as the component model of the sub-model library. Since the sub-training set after partitioning better distinguishes the complexity of the data, the output component model can better represent the characteristics of the data, and the single sub-training set further reduces the difficulty of calculation.

[0048] Step four: second screening of the surrogate model library. This stage mainly includes the following steps:

[0049] (1) Calculate the cross-validation error of each component model after the first screening:

[0050]

[0051] Wherein: is the predicted response of the i-th training point in the sub-training set by the component model.

[0052] (2) Set the cross-validation error threshold, remove the component models with cross-validation error higher than the threshold, and construct a new model library.

[0053] Step five: solve the adaptive weight coefficients of each component model and output the final model. This stage mainly includes the following steps:

[0054] (1) Use clustering method to process all basis functions, divide the design domain into O blocks, and obtain the corresponding basis functions of each block.

[0055] (2) The kernel probability density function of the i-th component model in the new model library at the test point x in the j-th block region (any training point in the j-th block region) is calculated as follows:

[0056]

[0057] Wherein: s 2 (x) is the local uncertainty of each component model in the new model library; is the predicted response of the i-th component model at the test point x in the j-th block region, y best_j (x) is the response of the basis function of the j-th block at the test point x, K is the number of component models after the second screening, K≤O. In formula (3), the Euclidean distance between the component model in the new model library and the basis function is selected as the basis for assigning the weight coefficient.

[0058] The local uncertainty of each component model in the new model library is calculated as follows:

[0059]

[0060] wherein: is the prediction standard deviation in Gaussian process, R is the covariance matrix between training points, and r is the covariance matrix between training points and test points.

[0061] (3) Determine the adaptive weight coefficients of each component model:

[0062]

[0063] wherein ω ij is the weight coefficient of the ith component model in the jth block region.

[0064] Combine each component surrogate model to output the final combined surrogate model:

[0065]

[0066] wherein, is the predicted response of the final combined surrogate model.

[0067] To test the actual performance of the present application, 37 test functions were respectively established on the same hardware and software platform to build T-AHS, PWS, BestPRESS, OWS full , OWS diag and the present application, a total of six combined surrogate models, for comparative study. The coefficient of determination R 2 , the normalized maximum absolute error NMAE (the maximum value of the absolute error between the normalized predicted response value and the true response value) and the normalized root mean square error NRMSE were used to evaluate the accuracy of each model, and a comprehensive investigation was carried out from the training amount (initial training set size) and test function dimension. Among them, R 2 and NRMSE are used to evaluate the global accuracy, and NMAE is used to evaluate the local accuracy. The number of initial training set training points is defined as 10ndv, wherein ndv is the dimension of the test function, and 1000ndv test points are randomly generated to calculate the accuracy evaluation index. Figures 2(a), 2(b) and 2(c) are the comparison results of overall performance (R 2 , NMAE and NRMSE three evaluation indexes), wherein the value of R 2 is closer to 1, and the values of NMAE and NRMSE are closer to 0, indicating that the prediction performance of the model is better. Considering the influence of samples with different number of training points on the prediction accuracy of the present application, four different sizes of initial training sets were selected, with sizes of 5ndv, 20ndv, ndv 2and (ndv+1)(ndv+2) / 2, FIG. 3(a), FIG. 3(b) and FIG. 3(c) are comparison results under different sizes of initial training sets, wherein the values of each evaluation index are the average values of the combined surrogate model under 37 test functions. FIG. 4 is a comparison of the accuracy of each model under different dimensions, 37 test functions are divided according to different dimensions, and the test results are shown in FIG. 4(a), FIG. 4(b) and FIG. 4(c), wherein the values of the evaluation index in FIG. 4(a) are the average values of the combined surrogate model under the same dimension test functions. As can be seen from the figures, the prediction accuracy of the present application is obviously better than that of other combined surrogate models, and better robustness is also shown under different sizes of initial training sets, and the overall accuracy and local accuracy are better in each dimension, indicating that the adaptive combined surrogate model based on fuzzy C-means clustering designed by the present application can provide more accurate, reliable and robust prediction results.

Claims

1. An adaptive combined surrogate modeling method based on fuzzy C-means clustering, characterized in that: Includes the following steps: Step 1: Generate a set of training points through Latin hypercube sampling, obtain the actual response of the design variables corresponding to the training points through simulation, and obtain an initial training set including training points and actual responses; use fuzzy C-means clustering to obtain the membership degree C of each training point to the cluster center, and divide the initial training set into O sub-training sets. Step 2: Build a proxy model library that includes multiple single proxy models; Step 3: Train each individual surrogate model in the surrogate model library using each sub-training set after the block is divided. After training each sub-training set, a sub-model library is obtained. Traverse O sub-training sets to obtain O sub-model libraries. Then, according to the prediction sum of squares criterion, calculate the global error of each individual surrogate model in the sub-model library, and select the individual surrogate model with the smallest global error in the sub-model library as the component model of that sub-model library. Step 4: Calculate the cross-validation error for each component model, set a cross-validation error threshold, and remove component models whose cross-validation errors exceed the threshold to form a new model library. Step 5: Use clustering to process all basis functions, divide the design domain into O blocks, obtain the corresponding basis functions for each block, determine the adaptive weight coefficients of each component model based on the Euclidean distance between the component models and basis functions in the new model library, and output the final combined surrogate model.

2. The adaptive combined surrogate modeling method based on fuzzy C-means clustering according to claim 1, characterized in that: The surrogate model library contains surrogate models constructed using the polynomial response surface methodology, radial basis function methodology, kriging methodology, and support vector regression methodology.

3. The adaptive combined surrogate modeling method based on fuzzy C-means clustering according to claim 2, characterized in that: The radial basis function method is divided into the quadratic radial basis function method and the thin plate spline radial basis function method.

4. The adaptive combined surrogate modeling method based on fuzzy C-means clustering according to claim 1, characterized in that: The formula for calculating the global error of a single surrogate model in the sub-model library is: Where: n is the number of training points in the sub-training set; y i This represents the true response of the i-th training point in the sub-training set; The predicted response of a single surrogate model trained on the set after deleting the i-th training point to the i-th training point in the sub-training set.

5. The adaptive combined surrogate modeling method based on fuzzy C-means clustering according to claim 4, characterized in that: The cross-validation error of the component model is calculated as follows: in: This represents the predicted response of the component model to the i-th training point in the sub-training set.

6. The adaptive combined surrogate modeling method based on fuzzy C-means clustering according to any one of claims 1 to 5, characterized in that: The final combined agent model: in, The final combined proxy model predicts the response; The weight coefficients of the i-th component model within the j-th region: The kernel probability density function of the i-th component model in the new model library at test point x in the j-th region is calculated as follows: in: Let y be the predicted response of the i-th component model at test point x in the j-th region. best_j (x) represents the response of the basis function of the j-th region at test point x, and K is the number of component models in the new model library, K≤O; The local uncertainty s of each component model in the new model library 2 (x) is calculated as follows: in: Let be the standard deviation of the prediction in the Gaussian process, R be the covariance matrix between training points, and r be the covariance matrix between training points and test points.

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