A fast evaluation method of surrogate model based on active subspace dimension reduction

Aimed at the problem of high-dimensional uncertainty quantification commonly existing in engineering complex shapes, traditional methods are often time-consuming and energy-consuming. So the paper proposes an efficient surrogate model method based on active subspace dimension reduction. The method uses active subspace to achieve feature dimension reduction and construct surrogate model, and develops a fast evaluation method of 1-D active subspace based on polynomial least squares decomposition to solve the difficult derivative problems. The detailed implementation process of the method is given in the paper, and the feasibility of the method is verified by examples such as the ablation thermal response and RAE2822 airfoil. It provides a solution for the quantification of multivariate uncertainty in engineering.


Introduction
As CFD plays an increasingly important role in industry, how to quickly quantify the impact of system instability caused by uncertainty has become an urgent need [1][2][3].Because the uncertainty in CFD numerical simulation presents significant high-dimensional feature [3], while the traditional MC simulation method [3] has high requirements on the number of samples and iterations of the solver, which limits its application in high-dimensional uncertainty quantification, there is an urgent need to develop more effective quantification methods.
In view of this situation, there are two mainstream approaches: The first is to use a more complex surrogate model [4][5] instead of solver operations, such as sparse learning [4], deep learning [5], etc; The second is to map high-dimensional data to low-dimensional space before calculating by using dimension reduction methods [6][7][8][9][10][11], such as PCA [6], sensitivity analysis [7], active subspace [8][9][10][11], etc.Compared with the construction of complex surrogate model, the cost of dimension reduction based modeling is relatively low, so the focus of this work is to build dimension reduction model.And compared with other dimension reduction methods, the dimension reduction process of active subspace method takes place in the eigenvector space rather than in the input feature space, which does not need clip the input data and retains the input integrity to the maximum extent.Hu [8] et al. combined global sensitivity method with active subspace to analyze mixed uncertainty parameters; Wang [9] et al. combined active subspace and response surface models to quantify the uncertainty of the turbulence model; Hu [10] et al. expressed the uncertainty effects affecting satellites in 1-D active subspace; Peng [11] proposed a reduced interval method for calculating active subspace eigenvectors and improved the computational efficiency by more than 95% by combining surrogate model.Although the above literature combines active subspace with surrogate model to quantify uncertainty, an efficient scheme is urgently needed due to the difficulty in derivative calculation in practical projects.
In this work, an efficient surrogate model method based on 1-D active subspace dimension reduction is proposed to meet the needs of rapid quantitative analysis of high-dimensional uncertainty in CFD applications.The examples of ablation thermal response and RAE2822 airfoil verify that the algorithm has the ability to solve the problem of multivariate uncertainty quantification in engineering, and provides a tool for rapid evaluation.

Active Subspace Dimension Reduction Model
Active subspaces use eigenvectors to construct low-dimensional spaces.Assuming that = represents the vector of all uncertainties, then the mean gradient function C corresponding to the system response f is represented by a symmetric and semi-positive matrix: In formula: W is an orthogonal matrix; is the mean of the product of the partial derivative of f , expressed as follows: In formula: Cij is the (ith, jth) element.For component i w ( 1, , i n   ) of matrix W , satisfy: From the above formula, the smaller the eigenvalue, the smaller the influence of the corresponding eigenvector on C .Descending on the eigenvalues, the transformed coordinates can be divided into two groups with greater and less influence according to the set thresholds, and their corresponding eigenvalues and eigenvectors are represented as: In formula: is the eigenvector of the first r larger eigenvalues, denoted as active subspace; The remaining eigenvector 2 W is recorded as an inactive subspace.It is difficult to solve the covariance matrix C directly in engineering, generally using MC method to sample M random samples and obtain the corresponding gradient value In formula: The eigenvector matrix Ŵ can be split into

A rapid uncertainty quantification method for 1-D active subspace dimension reduction
The above methods need to pay attention to the derivative information of output for all input uncertain variables at each sample point.It is difficult to calculate the derivative in practical projects.Therefore, a 1-D active subspace dimension reduction surrogate method based on polynomial least squares decomposition is proposed.As shown in Figure 1, the main implementation processes of the method are: 1) Complete linear polynomial regression analysis using existing sample points, that is: This polynomial can be solved by least squares Aα= b to get the coefficient 0 1 , ,  in the expansion.Each row of matrix A represents a sample point: In formula:   i j x represents the ith input parameter of the jth sample point.The output vector b is a vector consisting of the outputs of interest.
2) Calculate α by least squares, then extract elements from columns 2 to n+1 to build a new matrix   and normalize, like:       

Application Cases
In this section, we will use two application cases to verify the correctness and effectiveness of the proposed algorithm.

Case 1: ablation thermal response
Based on the active subspace dimension reduction method, we conduct a study on the uncertainty propagation of ablation thermal response.This example is a standard example provided by the official website of 4th Ablation Workshop [12], as shown in Figure 2: The temperature value at the 25mm thickness position is selected as the target output, and 9 material properties parameters are introduced as uncertainty inputs, as shown in Table 1: The 9 uncertain input parameters in Table 1 belong to typical cognitive uncertainties.Before uncertainty analysis, there is no prior knowledge about their distribution and importance, only the upper and lower bounds of the values are given, and the correlation between input parameters is not considered in the calculation process.Therefore, it is assumed that the parameters are uniformly distributed in the range of values.In this paper, 30 sets of parameter combinations generated by random sampling are used to calculate uncertain CFD to form sample data, and subsequent surrogate model construction is also based on this set of samples.
As shown in the figure below, the model proposed in this paper has been used to complete the comparison between the predicted value of the model and the CFD calculation results, and its determination coefficient R 2 is 0.9675, which meets the accuracy requirements and can be used for uncertainty analysis.As shown in figure 4, the model is resampled for 10000 times to obtain the histogram of temperature and fitted with gaussian curve.It can be seen from the figure that the temperature is in a normal distribution with an average of 355.396K and a standard deviation of 9.6143K.

Case 2: RAE2822 airfoil
Based on the above verification process, the following calculation example is the RAE2822 airfoil.This is a typical supercritical airfoil and a classic example to test the ability of CFD program to simulate transonic flow.Domestic and foreign research institutions have carried out a large number of wind tunnel tests and numerical simulation studies on this airfoil.The calculation state of this paper is:  .It is assumed that each parameter is uniformly distributed in its own support set, and its value range is shown in table 2. The following figure shows the comparison between the predicted value of the model in this paper and the CFD calculation result.From the figure, R 2 reaches 0.9955, which can prove that the model proposed in this paper can be used to replace the CFD calculation and complete the uncertainty measurement of multi-variables.

Conclusion
An efficient surrogate model method based on active subspace dimension reduction is proposed for the uncertainty measurement analysis of multi-input variables which is urgently needed in engineering applications.The feasibility of the method is verified by numerical examples such as the effect of the uncertainty of material properties on the prediction of ablation thermal response and prediction of lift coefficient under 9 uncertain inputs of the RAE2822 airfoil.This method provides a feasible solution to the problem of large dimension variable input uncertainty measurement for practical engineering problems.

. 3 ) 4 ) 5 )
According to the dimensionless matrix   , the projection y of all sample input vectors can be calculated: From y and b of all sample points, a quadratic response surface (QRS) surrogate model can be constructed: Calculate the fitting accuracy R 2 of 1-D active subspace.6)If the fitting accuracy meets the requirements, MC method is used to resample the one-dimensional and multi-dimensional polynomial and analyze the statistical information concerning the output.

Figure 3 .
Figure 3.Comparison of measured and predicted output quantity of interest.

Figure 4 .
Figure 4.The histogram of output quantity of interest.
concerned variable is the lift coefficient of the airfoil, and the original sample size is 100.The calculation grid is shown below.

6 Figure 5 .
Figure 5.The airfoil grid of RAE2822.The Spalart-Allmaras equation model is used for calculation, which is a turbulence model widely used in engineering.In use, the flow is assumed to be full turbulence, and the trip term in the model, which are: 1 2 2 3 1 3 4 , , , , , , , , b b w w v t t c c c c c c c  .It is assumed that each parameter is uniformly distributed in its own support set, and its value range is shown in table 2.

Figure 6 .
Figure 6.Comparison of measured and predicted output quantity of interest.

Table 2 .
The standard values and variation ranges of S-A model coefficients.