## 2 Proposed Multiple-Surrogate-Based Optimization Methodology  
### Multiple Surrogates  
The surrogate is a mathematical method to predict the unknown point's responses by building an approximate relationship between design variables and responses relying on limited sample point data information. Several expressions of popular surrogate models, including PRS, RBFNN, and KRG, are as follows:  
\[\text{PRS}:\ y_{\text{P}}=a_{0}+\sum\limits_{i=1}^{N}b_{i}x_{i}+\sum\limits_{i= 1}^{N}c_{ii}x_{i}^{2}+\sum\limits_{i=1}^{N-1}\sum\limits_{j=i+1}^{N}d_{ij}x_{i} x_{j}, \tag{1}\]  
where \(y_{\text{P}}\) is the output of PRS, \(x_{i}\) and \(x_{j}\) are the input variables, \(N\) is the number of input variables, and \(a_{0}\), \(b_{i}\), \(c_{ii}\), and \(d_{ij}\) are regression coefficients. \[\text{RBFNN}:\ y_{\text{r}}=\sum\limits_{i=1}^{M}\alpha_{i}\varphi(\|x-x_{i} \|,c), \tag{2}\]  
where \(y_{\text{r}}\) is the output of RBFNN, \(x\) is the input variable, \(M\) is the number of input variables, \(\alpha_{i}\) is coefficient, \(\|\ \|\) is the Euler norm, \(\varphi()\) is basis function, and spread constant, \(c\), is 0.5. \[\text{KRG}:\ y_{\text{k}}=\sum\limits_{i=1}^{K}\beta_{i}\phi_{i}(x)+z(x), \tag{3}\]  
where \(y_{\text{k}}\) is the output of KRG, \(x\) is the input variable, \(K\) is the number of input variables, \(\beta_{i}\) is coefficient, \(\phi_{i}()\) is basis function, and \(z()\) is the function of random error. The approximation accuracy of a surrogate is affected by sampling points. In other words, the best surrogate may be the worst one in the next iteration. And multiple surrogates are linearly superimposed by surrogate models. The best combination of surrogates for approximation can be gained by solving the coefficients during iterations, as follows:  
\[y(x)=w_{\text{P}}y_{\text{P}}(x)+w_{\text{r}}y_{\text{r}}(x)+w_{\text{k}}y_{ \text{k}}(x), \tag{4}\]  
where \(x\) is the design variables, \(y(x)\) is the output of multiple surrogate model, \(y_{\text{P}}(x)\), \(y_{\text{r}}(x)\), and \(y_{\text{k}}(x)\), respectively, are the output of PRS, RBFNN, and KRG, and \(w_{\text{P}}\), \(w_{\text{r}}\), and \(w_{\text{k}}\) are coefficients. The key to multiple surrogates is to solve coefficients \(w_{\text{P}}\),\(w_{\text{r}}\), and \(w_{\text{k}}\). [PERSON] et al. suggested that generalized mean square error (GMSE) is suitable for filtering out inaccurate surrogates [22], and it is as follows:  
\[GMSE=\frac{1}{n}\sum\limits_{i=1}^{n}\left(f_{i}-f_{\text{a}}\right)^{2}, \tag{5}\]  
where \(n\) is the number of sample points, and \(f_{i}\) and \(f_{\text{a}}\) represent the prediction output and actual output at \(x_{i}\), respectively. Using the GMSE to identify the approximation accuracy of the multiple surrogates, means that it is smaller, and the outputs of the multiple surrogates are closer to the real responses. Therefore, solving coefficients in Equation (4) can be converted into an optimization problem aimed at the smallest GRMS. The optimization problem is as follows:  
\[\left\{ \begin{array}{c}Find:w_{\text{P}},w_{\text{r}},w_{\text{k}}\\ Min:G=\frac{1}{\sum\limits_{i=1}^{L}}\left[y(x_{i})-y_{\text{a}}(x_{i}) \right]^{2}\end{array} \right., \tag{6}\]  
1. Determine the optimization problem, including objectives, constraints, and design variables, etc., and generate sample points of design variables as training points and testing points with the help of Latin hypercube sampling (LHS) [23]. LHS can more efficiently estimate the overall mean of response than estimation based on random sampling. 2. Perform the numerical simulation at the training points and testing points to obtain the interest responses. 3. Build the surrogates, including PRS, RBFNN, and KRG (reference for the method of construction [9; 20; 21]), respectively. With the minimum GMSE of the test points, solve the coefficients in Equation (10) and construct the multiple surrogates. 4. According to Equation (13), solve the determination coefficient \(R^{2}\). Check whether \(R^{2}\) is greater than 0.9.