## 2 RBFNN Approximation Method
### Mathematical Example  
Although it would be a special case that a high nonlinear problem such as mathematical example exists in an actual structural design problem, [PERSON]'s path function was considered as the mathematical example for the accuracy evaluation of RBFNN meta-model regarding to the SSP. Consider the following [PERSON]'s path function with two variables [31]:  
\[f(x)=-20 exp\left(-0.2\sqrt{\frac{x_{1}^{2}+x_{2}^{2}}{2}}\right)-exp\bigg{\{} \frac{cos[2\pi(x_{1})]+cos[2\pi(x_{2})]}{2}\bigg{\}}+20+e \tag{8}\]  
\[-2.0\leq x_{1},x_{2}\leq 2.0\]  
where \(f(x)\) is the mathematical function that presents a high nonlinearity with multimodal, and \(x_{1}\) and \(x_{2}\) are design variables. Both 200 training data and 30 testing data were randomly generated within the range of the design variables, for this function, and then the RBFNN based meta-models were fitted on 2-D and 3-D design spaces as shown in Figures 2 and 3.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Hyperparameter** & **Value** \\ \hline Learning rate & 0.7 \\ Number of training iteration & 10 \\ Number of hidden layers & 1 to 3 \\ Weight initialization & \(-1/25\), \(1/25\) \\ Spline smoothing parameter & 0.0 to 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Hyperparameter values.  
Figure 2: Meta-models on 2-D design space for [PERSON]’s path function.  
From Figures 2 and 3, it was clearly understood that the mathematical example has highly nonlinearity, and the RBFNN was apt for approximating such multimodal problem with large size training and testing data. As concerns the SSP, error analyses were carried out to evaluate the variation of performance according to the SSP values. The SSP values were varied from zero to 0.1 with 0.025 step size, and the accuracy measures of RBFNN meta-model such as RMSE and MAX were calculated using Equations (6) and (7). Their results were presented in Table 1 and Figure 4.  
As shown in Table 2 and Figure 4, the RMSE results were distributed from 0.154 to 0.218 according to the variation of SSP. The RMSE results were varied slightly until the SSP value was increased up to 0.05, whereas the RMSE values were increased significantly as the SSP value exceeded 0.05. The RMSE results also showed the inferior overall accuracy according to the increment of SSP. When approximating a design space with very high nonlinearity such as the mathematical example, very careful selection is necessary to apply the SSP value in order to guarantee the accuracy of RBFNN meta-model. As shown in Figure 4b, Local deviation was slightly changed except 0.075 of SSP value. The MAX value at 0.075 of SSP value was increased by 24% comparing to the result at 0.0 of SSP value. The abrupt increment of the MAX at 0.075 of SSP value was due to the highly nonlinearity of the [PERSON]'s path function.  
Figure 4: Performance comparison according to variation of SSP in [PERSON]’s path function.  
Figure 3: Meta-model on 3-D design space for [PERSON]’s path function.  
The characteristics of error scattering for the RBFNN meta-model were also evaluated according to the SSP values. The results of error scattering analyses were shown in Figure 5.  
Similar to the RMSE results, the error scattering results were varied slightly until the SSP value was increased up to 0.05, whereas the error scattering points deviated from the actual values were increased significantly as the SSP value exceeded 0.05. As shown in Figure 5d,e, the error scattering points were deviated widely according to the increment of SSP.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**SSP Value** & **RMSE** & **MAX** \\ \hline
0.0 & 0.1542 & 0.5198 \\
0.025 & 0.1594 & 0.5249 \\
0.050 & 0.1635 & 0.5172 \\
0.075 & 0.1876 & 0.6416 \\
0.1 & 0.2176 & 0.5316 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of accuracy according to variation of SSP in [PERSON]’s path function.  
Figure 5: Error scattering analysis according to variation of SSP in [PERSON]’s path function.