## 2 ARBF-MLPA Method
### Multi-Objective Optimization Algorithm
Step 3: obtain the values of the objective function and constraint function at the initial sampling point. In practical engineering problems, the values of objective function and constraint function are usually obtained using finite element simulation calculation, so this step is a relatively time-consuming process. Step 4: build a radial basis function neural network model based on the sample points obtained in step 3 and relevant response values. Step 5: in the metamodel obtained in step 4, solve the multi-objective problem applying NSGA-II algorithm to obtain the Pareto optimal front edge, and normalize the obtained response value of Pareto front edge in iteration to improve the accuracy of the model. Step 6: add the high-quality point on the optimal front edge obtained through ARBF-MLPA to the model to be built next time, as training sample, these points will lead the optimization results to be closer to the real front edge. As shown in Figure 2, the sample points added in iteration each time are uniformly distributed between neighboring design sample points which are farther from Pareto front edge. As shown in scenario 1 in Figure 2, there are two large gaps between the design points distributed in the Pareto front, and we can uniformly select points between the two gaps as sample points to add to the model. As shown in scenario 2 in Figure 2, there are four design points in the Pareto front, but the gap between the two points is large, so we can uniformly select points in the middle of the large gap as sample points. Different from the traditional uniform point-adding method, this method is purposeful and selective in a certain design sample gap, rather than blindly uniform adding points. Figure 1: Flow chart of NSGA-II algorithm. Step 7: select the extreme points in the Pareto optimal front edge corresponding to each target. The number of extreme points should be equal to the number of targets. This paper is presenting a two-target problem, and two extreme points should be selected. Step 8: in the Pareto approximate optimal front edge, select the (Min) point set which is the nearest to the existing training sample, then select (Max) point which has greater relative distance from these point sets. As is generally the case, the model accuracy will be very low in sparse regions since MMD is exactly in such regions. So, these points are selected and taken as a new training sample, and this helps improve the local accuracy of models [21; 22; 23]. MMD points of Pareto front can be selected using the equation below:  
\[\max\Bigl{[}\min_{1\leq i\leq s_{i},1\leq j\leq t}\bigl{(}d\left(x_{i},x_{j} \right)\bigr{)}\Bigr{]} \tag{2}\]  
where, \(x_{i}\) and \(x_{j}\) respectively represent sample \(i\) and sample \(j\) of Pareto front point pool (\(S\)) and sample pool (\(P\)) in the last iteration; \(d\left(x_{i},x_{j}\right)\) represents Euclidean distance between \(x_{i}\) and \(x_{j}\). Figure 2: Pick extra sample points in Pareto points. (**a**) Scenario 1. (**b**) Scenario 2. Step 9: search for the Nearest Boundary (NB) points from each MMD point sample (obtained in step 7) and generate MMD-NB point, using the equation below:  
\[\mathrm{x}_{MMD-NB}=\left(\frac{\mathrm{x}_{MMD}+\mathrm{x}_{NB}^{EP}}{\alpha_{1 }}\right)\beta_{1}+\left(\frac{\mathrm{x}_{MMD}+\mathrm{x}_{NB}}{\alpha_{2}} \right)\beta_{2} \tag{3}\]  
wherein, \(\mathrm{x}_{NB}^{EP}\) represents the extreme point that satisfies the closest Euclidean distance to the design feasible region, and \(\mathrm{x}_{NB}\) represents the corresponding NB sample points; \(\mathrm{x}_{MMD}\) represents MMD sample point, \(\mathrm{x}_{MMD-NB}\) represents the obtained MMD-NB point; \(\alpha_{i}\) represents the number of designed variables, which is 2 in this paper, \(\beta_{i}\) represents threshold value, which is set to be 0.5 here, Figure 3 below shows the distribution of MMD-NB sample points in 2D space.