CoKriging modeling method based on Hessian enhancement

Abstract

The invention discloses a CoKriging (Hessian-enhanced Cokriging, HECoK) modeling method based on Hessian enhancement, and belongs to the field of multi-fidelity modeling. The method comprises the following steps: acquiring a small amount of high-precision sample data including a function value, a first-order gradient and a second-order gradient; acquiring a large amount of low-precision sample data containing the same information; the method comprises the following steps: expanding an existing gradient enhancement CoKriging algorithm, and introducing second-order gradient information; and constructing a Hessian enhanced CoKriging model based on the high-precision data and the low-precision data. According to the method, under the condition of scarcity of high-precision sample data, the modeling precision can be remarkably improved by fusing first-order and second-order gradient information, and the problem of prediction performance reduction caused by insufficient high-precision information in traditional multi-fidelity modeling is effectively relieved.

Description

Technical Field

[0001] The present invention belongs to the field of multi-fidelity modeling, and in particular relates to a CoKriging modeling method based on Hessian enhancement. Background Art

[0002] CoKriging, a classic and theoretically mature multi-fidelity modeling method, has garnered widespread attention. An extension of Kriging, CoKriging establishes a joint covariance structure between primary variables (high-precision data) and auxiliary variables (low-precision data). This allows the model to approximate the exact values ​​of high-precision data points while also leveraging the trend information from a large amount of low-precision data for global modeling. This significantly reduces the need for expensive sample points while maintaining model accuracy. This model was originally developed in the field of geostatistics and has since been incorporated into aerodynamic research by many researchers. The autoregressive multi-fidelity model (KOH model) proposed by Kennedy and O'Hagan (Kennedy MC, O'Hagan A. Predicting the output from a complex computer code when fast approximations are available [J]. Biometrika, 2000, 87 (1): 1-13.) provides a theoretical basis for CoKriging. Forrester et al. (Kuya, Y., Takeda, K., Zhang, X., & Forrester, AI (2011). Multifidelity surrogate modeling of experimental and computational aerodynamic datasets. AIAA journal, 49 (2), 289-298.) successfully applied it to wing design optimization, verifying its effectiveness in engineering problems.

[0003] In addition to using inexpensive, low-precision data as auxiliary information to improve model prediction accuracy, gradient information is equally effective (or even more effective). To further improve modeling accuracy, the gradient-enhanced CoKriging (GECoK) method has been proposed and has gradually attracted attention. This method introduces first-order gradient information on the basis of CoKriging. Compared with models that rely solely on function values, GECoK can more effectively reflect the local variation trend of the response function in the design space, thereby improving modeling accuracy. GECoK proposed by Deng et al. (Deng, Yixiang, Guang Lin, and Xiu Yang. "Multifidelity data fusion via gradient-enhanced Gaussian process regression." arXiv preprint arXiv:2008.01066(2020).) extends the KOH model and can not only predict high-fidelity function values, but also predict their gradient information.

[0004] Currently, most gradient boosting models focus on utilizing first-order gradient information, while the potential of second-order gradient information has not been fully explored. Therefore, the main contribution of this invention is to further extend the existing gradient boosting CoKriging method to incorporate second-order gradient information, further improving the prediction accuracy of multi-fidelity models. Summary of the Invention

[0005] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is as follows:

[0006] A CoKriging modeling method based on Hessian enhancement includes the following steps:

[0007] S1. Obtain a small amount of high-precision sample data containing independent variables, function values, first-order gradients, and second-order gradient information.

[0008] S2. Obtain a large amount of low-precision sample data containing the same information

[0009] S3. Constructing a low-fidelity Hessian enhanced Kriging model HEK using low-precision sample data l ;

[0010] S4. Constructing the Hessian enhanced Kriging difference model HEK using the residuals of high and low precision sample data d ;

[0011] S5. Build the Hessian enhanced CoKriging model HECoK and use the model to predict new data.

[0012] The present invention provides a CoKriging modeling method based on Hessian enhancement. This method addresses the existing problems of low-precision data being abundant but inaccurate, and high-precision data being limited, resulting in insufficient modeling accuracy. By incorporating first- and second-order gradient information from both high- and low-precision sample data into the modeling process, the method effectively improves the model's prediction accuracy even when high-precision data is scarce. By fully utilizing this multi-source gradient information, the model's prediction accuracy is effectively improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings used in the embodiments.

[0014] Figure 1 is a framework diagram of the Hessian enhanced CoKriging (HECoK) model described in the embodiments;

[0015] Figure 2 Figures (a) to (d) show the prediction results of the analytic function using the gradient-enhanced CoKriging (GECoK) and Hessian-enhanced CoKriging (HECoK) methods, respectively, and a comparison of the absolute errors of the prediction results of the two methods (e). DETAILED DESCRIPTION

[0016] To make the technical solution of the present invention clearer, the present invention will be clearly and completely explained below with reference to the accompanying drawings of the embodiments of the invention. The embodiments described only use a part of the embodiments, not all of the embodiments. The present invention is specifically implemented according to the following steps:

[0017] S1. Obtain a small amount of high-precision sample data containing independent variables, function values, first-order gradients, and second-order gradient information.

[0018] S2. Obtain a large amount of low-precision sample data containing the same information

[0019] In the embodiment of the present invention, an analytical function is used to sample 3 and 6 high-precision and low-precision data respectively, and valid first-order and second-order gradient information is included, and the sampling method is uniform sampling.

[0020] S3. Based on low-precision sample data Construct Hessian enhanced Kriging model HEK l .

[0021] The Hessian enhanced Kriging model in the present invention constructs the covariance matrix R through the kernel function R(·,·):

[0022]

[0023] in,

[0024]

[0025] At the new sample position x * The function value prediction formula at is as follows:

[0026]

[0027] in,

[0028]

[0029] The prediction of the second-order gradient is as follows:

[0030]

[0031] in,

[0032]

[0033] The hyperparameter θ in formula (4) is optimized by minimizing the following likelihood function through genetic algorithm:

[0034]

[0035] in, represents the estimated variance.

[0036] S4. Residuals based on high and low precision sample data Construct Hessian enhanced Kriging difference model HEK d ,The model construction process is similar to the S3 described above.

[0037] S5. HEK-based model l and HEK d , build the fusion model HECoK.

[0038] The covariance matrix of HECoK is as follows:

[0039]

[0040] Among them, R l 、R d The model HEK l and HEK d The covariance matrix constructed by the kernel function in . The hyperparameters θ in the kernel function of these two modelsl ,θ d The regression parameter ρ is optimized by the likelihood function of Equation (9). At the new sample position, the HECoK prediction formula for the posterior mean is the same as Equation (5), and the corresponding matrix or vector contains the second-order gradient information.

[0041] In the embodiment provided by the present invention, the function value, first-order gradient, and second-order gradient in the high-precision sample data are obtained from equations (11), (12), and (13), respectively:

[0042] y h =(6x-2) 2 sin(12x-4),x∈[0,1] (11)

[0043] y′ h =12(6x-2)sin(12x-4)+12(6x-2) 2 cos(12x-4),x∈[0,1] (12)

[0044] y′ h ′=[72-144(6x-2) 2 ]sin(12x-4)+288(6x-2)cos(12x-4),x∈[0,1] (13)

[0045] The low-precision sample data is obtained through equations (14), (15), and (16):

[0046] y l =0.5y h +10(x-0.5)-5,x∈[0,1] (14)

[0047] y l ′=6(6x-2)sin(12x-4)+6(6x-2) 2 cos(12x-4)+10,x∈[0,1] (15)

[0048] y l = [36-72(6x-2) 2 ]sin(12x-4)+144(6x-2)cos(12x-4),x∈[0,1] (16)

[0049] The sampling positions of high and low precision data are x h =(0,0.4,1),x l=(0,0.2,0.4,0.6,0.8,1). In order to verify the effectiveness of the CoKriging method based on Hessian enhancement in this invention, the gradient enhancement CoKriging method is used as a comparison method. When the amount of data is the same, the two methods respectively predict the function value as follows: Figure 2 As shown in (a) and (c), Figure 2 (e) shows the absolute error (AE) comparison of the prediction results of the two methods. It can be seen that the method of the present invention significantly improves the prediction accuracy, proving its advantages in the field of multi-fidelity modeling. In addition, this embodiment also provides two methods for predicting the first-order gradient and the second-order gradient, such as Figure 2 As shown in (b) and (d).

[0050] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may be modified and varied in various ways.

Claims

1. A CoKriging modeling method based on Hessian enhancement, characterized in that: The steps include: S1. Obtain a small amount of high-precision sample data containing independent variables, function values, first-order gradients, and second-order gradient information. S2. Obtain a large amount of low-precision sample data containing the same information S3. Constructing a low-fidelity Hessian enhanced Kriging model HEK using low-precision sample data l ; S4. Constructing the Hessian enhanced Kriging difference model HEK using the difference between high and low precision sample data d ; S5. Build the Hessian enhanced CoKriging (HECoK) model and use it to predict new data.

2. A CoKriging modeling method based on Hessian enhancement according to claim 1, characterized in that: The high-precision sample data It comes from the high-fidelity function in the analytical function and its first and second-order derivative functions.

3. A CoKriging modeling method based on Hessian enhancement according to claim 1, characterized in that: The low-precision sample data It comes from the low-fidelity function in the analytical function and its first and second-order derivative functions.

4. The CoKriging modeling method based on Hessian enhancement according to claim 1, characterized in that: The low-fidelity Hessian enhanced Kriging model HEK is constructed using low-precision sample data l , specifically including: Based on the Hessian-enhanced Kriging (HEK) method, using low-precision sample data Building a low-fidelity model HEK l .

5. The CoKriging modeling method based on Hessian enhancement according to claim 1, characterized in that: The residuals of high and low precision sample data are used to construct the Hessian enhanced Kriging difference model HEK d , specifically including: Based on the Hessian enhanced Kriging method, the difference model HEK is constructed using the residuals of high and low precision sample data. d , the residual is: Where ρ is the associated HEK l and HEK d The regression parameters of the two models are HEK l At the high-fidelity sample position x h The predicted value of the function at and are the corresponding first-order and second-order gradient prediction values ​​respectively.

6. The CoKriging modeling method based on Hessian enhancement according to claim 1, characterized in that: The Hessian Enhanced CoKriging (HECoK) model is constructed and used to predict new data, specifically including: Based on the HEK model l and HEK d , build the fusion model HECoK: HECoK=ρHEK l +GATE d (2) The parameter ρ can be optimized by genetic algorithm during model training. The prediction of new data is calculated by the prediction formula of HECoK model.

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