CN113868853A - Gradient enhancement variable fidelity proxy model modeling method - Google Patents

Gradient enhancement variable fidelity proxy model modeling method Download PDF

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CN113868853A
CN113868853A CN202111105898.5A CN202111105898A CN113868853A CN 113868853 A CN113868853 A CN 113868853A CN 202111105898 A CN202111105898 A CN 202111105898A CN 113868853 A CN113868853 A CN 113868853A
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宋学官
李昆鹏
王硕
来孝楠
何西旺
杨亮亮
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Dalian University of Technology
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Abstract

The invention discloses a gradient enhancement variable fidelity proxy model modeling method, and belongs to the technical field of proxy models. The method comprises the following steps: (1) and establishing a low-fidelity model according to the low-fidelity sample and the corresponding low-fidelity real response and low-fidelity real gradient thereof. Taking the gradient enhanced radial basis function model as a low-fidelity model, and giving a low-fidelity prediction response and a low-fidelity prediction gradient at a high-fidelity sample through the established low-fidelity model; (2) introducing a scale factor and a difference function to correct the low-fidelity model; (3) solving the column vector of undetermined coefficients in the scaling factor and the difference function; (4) and establishing a prediction expression of the gradient enhancement variable fidelity proxy model. The method can fully consider the physical significance of the gradient information on the basis of the traditional variable fidelity agent model, thereby further enhancing the prediction precision of the model. Meanwhile, Cholesky decomposition is adopted when the established correlation matrix is inverted, and the calculation efficiency can be effectively ensured.

Description

Gradient enhancement variable fidelity proxy model modeling method
Technical Field
The invention belongs to the technical field of proxy models, and relates to a gradient enhancement variable fidelity proxy model modeling method.
Background
The agent model establishes a mathematical approximation for the relationship between design variables and complex system responses, is focused on solving the problems that the calculation amount is too large or the existing calculation resources cannot meet the calculation requirements in engineering design, and has been widely concerned and applied in different scientific fields (such as design optimization, material design, reliability analysis, digital twinning, uncertainty quantification and the like). In addition, the proxy model can provide powerful help in processing noisy or missing data and in understanding the complex functional relationships between design variables and output responses.
Although the proxy model method saves a lot of simulation computation resources and high experiment cost in engineering optimization design, the sample response for constructing the proxy model usually comes from high fidelity numerical simulation or high precision actual experiment, which usually also needs to consume expensive cost. In addition, with the increase of design variable dimensions in the engineering practical optimization problem, the number of times of required simulation or experiment operation is correspondingly increased, which prevents the agent model from being popularized and used in the engineering optimization problem and brings great challenges to further development of the agent model technology. To further reduce the computational or experimental costs required to build a proxy model, researchers have proposed a variable fidelity proxy model. Compared with the common agent model method, the variable fidelity agent model organically fuses information contained in high fidelity samples and low fidelity samples, and the main idea is to use a large number of low fidelity samples as assistance, so that the agent model with satisfactory precision can be constructed by using only a small number of high fidelity samples. This provides a very attractive solution for further improving the efficiency of constructing the proxy model and reducing the required computation time and resources, and thus becomes one of the research hotspots in the field of proxy models. In addition, according to the existing research, if the gradient can be obtained, the gradient can be used as auxiliary information to further improve the performance of the variable fidelity proxy model. The physical meaning of the gradient reflects the change rate, and the change trend of the function can be fully reflected, which means that the performance of the proxy model can be further improved if the gradient of the approximate function established by the proxy model can be ensured to be consistent with the gradient of the real function. However, the current gradient enhancement variable fidelity proxy model only uses gradient information as a sample response to improve the accuracy of the model, and does not fully consider the physical significance. In addition, the introduction of gradient information causes the dimension of a correlation matrix in a model to be greatly increased, and the modeling efficiency is seriously reduced. Therefore, in order to further improve the prediction accuracy of the gradient enhancement fidelity proxy model on the premise of ensuring the modeling efficiency, it is necessary to provide a gradient enhancement fidelity proxy model that sufficiently considers the physical significance of the gradient.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a modeling method of a gradient enhancement variable fidelity proxy model, which can fully consider the physical significance of gradient information on the basis of the traditional variable fidelity proxy model, thereby further enhancing the prediction precision of the model. Meanwhile, Cholesky decomposition is adopted when the established correlation matrix is inverted, and the calculation efficiency can be effectively ensured.
The technical scheme of the invention is as follows:
a gradient enhancement variable fidelity proxy model modeling method comprises the following steps:
(1) and establishing a low-fidelity model according to the low-fidelity sample and the corresponding low-fidelity real response and low-fidelity real gradient thereof. Taking the gradient enhanced radial basis function model as a low-fidelity model, and giving a low-fidelity prediction response and a low-fidelity prediction gradient at a high-fidelity sample through the established low-fidelity model;
(2) introducing a scale factor and a difference function to correct the low-fidelity model;
(3) solving the column vector of undetermined coefficients in the scaling factor and the difference function;
(4) and establishing a prediction expression of the gradient enhancement variable fidelity proxy model.
Further, in the step (1), the low-fidelity prediction model established can provide not only a low-fidelity prediction response at high-fidelity samples, but also a low-fidelity prediction gradient at high-fidelity samples.
Further, in the step (3), when solving the scaling factor and the array vector of the to-be-determined coefficient in the difference function, the gradient enhancement variable fidelity proxy model designed by the invention adopts Cholesky decomposition to ensure the calculation efficiency when performing matrix inversion.
The invention has the beneficial effects that: the invention provides a modeling method based on a gradient enhancement variable fidelity proxy model, which can fully consider the physical significance of gradient information on the basis of the traditional variable fidelity proxy model, thereby further enhancing the prediction accuracy of the model. Meanwhile, Cholesky decomposition is adopted when the established correlation matrix is inverted, and the calculation efficiency can be effectively ensured.
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FIG. 1 is a schematic flow chart of a gradient enhancement variable fidelity proxy model modeling method according to the present invention.
FIG. 2 is a graph of average ranking comparison of prediction accuracy for the present invention and other methods.
Detailed Description
The invention is further illustrated by the following description in conjunction with the accompanying drawings and specific examples.
In order to keep the generality, n high-fidelity samples are provided
Figure BDA0003272335380000021
The method comprises the steps of obtaining a high-fidelity real response and a high-fidelity real gradient corresponding to an ith high-fidelity sample, wherein i is 1, 2. There are p low fidelity samples xL＝{xl1，xl2，...，xlp}，
Figure BDA0003272335380000022
Is the high fidelity true response and high fidelity true gradient corresponding to the jth low fidelity sample. x is the number ofhi(i ═ 1,2,. n), and xljThe dimension of (j ═ 1, 2.... times.p) is d dimension. The invention designs a gradient increasing deviceA method for modeling a strongly varying fidelity proxy model, as shown in fig. 1, the method comprising the steps of:
(1) and establishing a low-fidelity model according to the low-fidelity sample and the corresponding low-fidelity real response and low-fidelity real gradient thereof. The gradient-enhanced radial basis function model is used as a low-fidelity model, the expression of the model is shown in formula (1), and a low-fidelity prediction response and a low-fidelity prediction gradient of a high-fidelity sample are given through the established low-fidelity model, wherein the low-fidelity prediction response and the low-fidelity prediction gradient of a kth (k is 1, 2.. the., n) high-fidelity sample are as follows:
Figure BDA0003272335380000031
wherein
Figure BDA0003272335380000032
Is a quadratic differentiable Multi-quadric radial basis function; | represents the euclidean distance; beta is aiAnd
Figure BDA0003272335380000033
the model parameters can be obtained by a least square method; superscript e denotes sample dimension, e ═ 1, 2.. d;
Figure BDA0003272335380000034
is composed of high fidelity samples xhkLow fidelity predictive response of
Figure BDA0003272335380000035
And low fidelity prediction gradient
Figure BDA0003272335380000036
A column vector of components.
(2) And introducing a scale factor and a difference function to correct the low-fidelity model. Using a scale factor p and an extended correlation matrix
Figure BDA00032723353800000318
And the column direction consisting of undetermined coefficientsThe quantity ω corrects the low fidelity model described in step (1):
Figure BDA0003272335380000037
wherein the content of the first and second substances,
Figure BDA0003272335380000038
is a difference function; f. ofH(xH) Is a column vector containing high fidelity true response and high fidelity true gradient;
Figure BDA0003272335380000039
is a column vector containing a low-fidelity prediction response and a low-fidelity prediction gradient, which are given by the low-fidelity model described in step (1). Extended correlation matrix between high fidelity samples
Figure BDA00032723353800000310
It can be written in the form of a block matrix as an n × n sub-matrix:
Figure BDA00032723353800000311
wherein the content of the first and second substances,
Figure BDA00032723353800000312
sub-matrix phi inhc，hm(c ═ 1,2,. n, m ═ 1,2,. n) can be expressed as
Figure BDA00032723353800000313
(3) And solving the scaling factor p and the undetermined coefficient column vector omega in the difference function. By constructing an integrated matrix
Figure BDA00032723353800000314
Equation (2) can be rewritten into a more compact matrix formEquation (5) to solve for the scale factor ρ and the undetermined coefficient column vector ω.
Figure BDA00032723353800000315
Wherein the content of the first and second substances,
Figure BDA00032723353800000316
is an augmented column vector consisting of a scale factor p and a column vector omega of undetermined coefficients. According to the least square method, can be solved
Figure BDA00032723353800000317
Wherein the content of the first and second substances,
Figure BDA0003272335380000041
the first element of (a) is the scale factor p and the remaining elements are the column vectors ω.
Solving as described in step (3)
Figure BDA0003272335380000042
The gradient enhancement fidelity model provided by the invention can ensure that the prediction trend at a high-fidelity sample is strictly consistent with the high-fidelity real trend, so that the aim of improving the prediction precision is fulfilled.
In addition, due to the introduction of gradient information, a matrix is caused
Figure BDA0003272335380000043
Is large, so that the solution is carried out
Figure BDA0003272335380000044
Time of day calculation
Figure BDA0003272335380000045
The reverse efficiency of (c) is too low. The invention adopts Cholesky decomposition to divide the matrix
Figure BDA0003272335380000046
And decomposing, thereby ensuring the calculation efficiency in inversion.
(4) And establishing a prediction expression of the gradient enhancement variable fidelity proxy model. In obtaining
Figure BDA0003272335380000047
Then, the prediction expression of the gradient enhancement variable fidelity agent model designed by the invention on any x in the space to which the sample belongs is as follows:
Figure BDA0003272335380000048
wherein the content of the first and second substances,
Figure BDA0003272335380000049
high fidelity prediction response is given to the gradient enhancement variable fidelity agent model;
Figure BDA00032723353800000410
Figure BDA00032723353800000411
is a column vector whose elements are the low-fidelity prediction response and the low-fidelity prediction gradient over x from the low-fidelity model described in step (1);
Figure BDA00032723353800000412
is an extended correlation matrix between x and high fidelity samples, of the form:
Figure BDA00032723353800000413
wherein the sub-matrix phix，hv(v ═ 1, 2.. times, n) is
Figure BDA00032723353800000414
To test the actual performance of the method, the coefficient R was determined on the same software and hardware platform2For evaluation criteria, for the data from https: html 10 test functions of/www.sfu.ca/-ssurjano/index. html comparative experiments of variable fidelity radial basis function (MFS-RBF), cooperative kriging (CoKRG), Gradient Enhanced Kriging (GEKRG) and Gradient Enhanced Radial Basis Function (GERBF) were performed. FIG. 2 shows the comparison result of the average ranking of prediction accuracy between the present invention and other methods, wherein the abscissa represents each agent model method, and the ordinate represents the ranking value of the corresponding agent model method, and the lower the ranking value, the better the agent model method. As can be seen from the figure, the average ranking of the invention is superior to the average ranking of other agent models, which shows that the gradient enhancement variable fidelity agent model modeling method designed by the invention can provide more accurate prediction results.

Claims (1)

1. A gradient enhancement variable fidelity proxy model modeling method is characterized by comprising the following specific steps:
is provided with n high-fidelity samples
Figure FDA0003272335370000011
The high fidelity real response and the high fidelity real gradient corresponding to the ith high fidelity sample are shown, wherein i is 1,2, …, n; there are p low fidelity samples xL＝{xl1,xl2,…,xlp}，
Figure FDA0003272335370000012
The high fidelity true response and the high fidelity true gradient corresponding to the jth low fidelity sample; x is the number ofhi(i ═ 1,2, …, n) and xljThe dimension of (j ═ 1,2, …, p) is d dimension:
the method comprises the following specific steps:
(1) establishing a low-fidelity model according to the low-fidelity sample and the corresponding low-fidelity real response and low-fidelity real gradient thereof: taking the gradient enhanced radial basis function model as a low-fidelity model, wherein the model expression is shown in formula (1), and giving a low-fidelity prediction response and a low-fidelity prediction gradient at a high-fidelity sample through the established low-fidelity model, wherein the low-fidelity prediction response and the low-fidelity prediction gradient of the kth high-fidelity sample are as follows:
Figure FDA0003272335370000013
wherein k is 1,2, …, n,
Figure FDA0003272335370000014
is a quadratic differentiable Multi-quadric radial basis function; | represents the euclidean distance; beta is aiAnd
Figure FDA0003272335370000015
the model parameters are obtained by a least square method; superscript e denotes sample dimension, e ═ 1,2, …, d;
Figure FDA0003272335370000016
is composed of high fidelity samples xhkLow fidelity predictive response of
Figure FDA0003272335370000017
And low fidelity prediction gradient
Figure FDA0003272335370000018
A column vector of components;
(2) and (3) introducing a scale factor and a difference function to correct the low-fidelity model: using a scale factor p and an extended correlation matrix
Figure FDA0003272335370000019
And correcting the low-fidelity model in the step (1) by a column vector omega consisting of the coefficients to be determined:
Figure FDA00032723353700000110
wherein the content of the first and second substances,
Figure FDA00032723353700000111
is a difference function; f. ofH(xH) Is a column vector containing high fidelity true response and high fidelity true gradient;
Figure FDA00032723353700000112
is a column vector comprising a low-fidelity prediction response and a low-fidelity prediction gradient, wherein the low-fidelity prediction response and the low-fidelity prediction gradient are given by the low-fidelity model described in step (1); extended correlation matrix between high fidelity samples
Figure FDA00032723353700000113
It can be written in the form of a block matrix as an n × n sub-matrix:
Figure FDA00032723353700000114
wherein the content of the first and second substances,
Figure FDA00032723353700000115
sub-matrix phi inhc,hm(c ═ 1,2, … n, m ═ 1,2, … n) can be represented by
Figure FDA0003272335370000021
(3) Solving the proportional factor rho and a series vector omega of the coefficient to be determined in the difference function; by constructing an integrated matrix
Figure FDA0003272335370000022
Equation (2) is rewritten to a more compact matrix form, see equation (5), to facilitate the resolution of the scaling factorA sub rho and a pending coefficient column vector omega;
Figure FDA0003272335370000023
wherein the content of the first and second substances,
Figure FDA0003272335370000024
the column vector is an augmented column vector consisting of a scale factor rho and a column vector omega of a pending coefficient; according to the least square method, can be solved
Figure FDA0003272335370000025
Wherein the content of the first and second substances,
Figure FDA0003272335370000026
the first element of (a) is a scale factor ρ, and the remaining elements are column vectors ω; using Cholesky decomposition to divide the matrix
Figure FDA0003272335370000027
Decomposing to ensure the calculation efficiency of solving the reverse time;
(4) establishing a prediction expression of a gradient enhancement variable fidelity proxy model: in obtaining
Figure FDA0003272335370000028
Then, the prediction expression of the gradient enhancement variable fidelity proxy model to any x in the space to which the sample belongs is as follows:
Figure FDA0003272335370000029
wherein the content of the first and second substances,
Figure FDA00032723353700000210
high fidelity prediction response is given to the gradient enhancement variable fidelity agent model;
Figure FDA00032723353700000211
is a column vector, the elements are the low-fidelity prediction response and the low-fidelity prediction gradient on x by the low-fidelity model described in step (1);
Figure FDA00032723353700000212
is an extended correlation matrix between x and high fidelity samples, of the form:
Figure FDA00032723353700000213
wherein the sub-matrix phix,hv(v-1, 2, …, n) is
Figure FDA00032723353700000214
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Publication number Priority date Publication date Assignee Title
CN117077551A (en) * 2023-10-17 2023-11-17 中国科学院工程热物理研究所 Non-embedded uncertainty quantization method based on gradient enhancement stochastic Co-Kriging model
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