## 3 Improved PSO-RBFNN Model
### Improved PSO-RBFNN  
This section describes the improved PSO-RBFNN that can tune network parameters during the training process for higher accuracy. The RBFNN parameters optimization problem can be described as follows:  
\[find\ \mathbf{z}_{op}=\underset{\mathbf{z}}{\text{argmax}}g(\mathbf{z}) \tag{11}\] \[s.t.\ \mathbf{z}\in\mathbf{\Re}^{D}\]  
where \(\mathbf{z}=(\sigma_{1},\ldots,\sigma_{N},w_{11},\ldots,w_{mN},\xi_{1},\ldots,\xi_ {m})\) is the position of a particle, with \(\sigma_{i}\), \(w_{ji}\), and \(\xi_{j}\) being the width, connection weights, and residual sequence, respectively. The dimension of a particle, \(D\), is determined by  
\[D=N\times m+N+m \tag{12}\]  
In the improved PSO-RBFNN, the fitness value of each particle represents the accuracy of the network. The fitness value is given by  
\[g(\mathbf{z})=\frac{1}{1+MSE(\mathbf{y}_{r}(\mathbf{z}),\mathbf{y}_{e}(\mathbf{z}))} \tag{13}\]  
where \(g(\mathbf{z})\) is the scalar fitness function; \(\mathbf{y}_{r}(\mathbf{z})\) and \(\mathbf{y}_{e}(\mathbf{z})\) are the network real output and expected output, respectively; \(MSE\) is the mean square error function referring to [34]. A larger fitness value implies a smaller \(MSE\) value. Based on Equations (5) and (9), the improved PSO is used to find the optimal particle \(\mathbf{z}_{op}=(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{N},w^{\prime}_{11}, \ldots,w^{\prime}_{mN},\xi^{\prime}_{1},\ldots,\xi^{\prime}_{m})\), and then determine the optimal RBFNN. Figure 5 shows the schematic optimization process for obtaining \(\mathbf{z}_{op}\), and Figure 6 summarizes the procedure for the improved PSO-RBFNN.  
Figure 4: Schematic Optimization process of our improved particle swarm optimization (PSO).  
Figure 5: The schematic optimization process for obtaining \(\mathbf{z}_{op}\).  
Fast Multi-Objective Antenna Optimization Framework Combining MOEAs and Improved PSO-RBFNN Surrogate Model  
In this section, we use the improved PSO-RBFNN surrogate model discussed in the previous section, rather than conventionally time-consuming EM simulations, to evaluate the antenna performance. The constructed antenna surrogate model is a black box for mapping the relationship between the antenna structure parameters and performance indexes (e.g., reflection coefficients, gain, efficiency, etc.). At the same time, the antenna multi-objective optimization is carried out by MOEAs. The whole optimization framework can be summarized as follows:  
1. Predefine antenna geometry vector \(\mathbf{x}\) and design space \(\mathbf{X}\);
2. Obtain the sample set \(\mathbf{S}\) by sampling randomly in the design space \(\mathbf{X}\) and obtain the response set \(\mathbf{Y}\) by calling for EM simulation software;
3. Obtain the optimal RBFNN parameters \(\mathbf{z_{\mathrm{op}}}\) using improved PSO based on \(\mathbf{S}\), \(\mathbf{Y}\);
4. Construct the improved PSO-RBFNN model \(\mathbf{R}_{s}(\mathbf{x})\);
5. Optimize the population by MOEAs and \(\mathbf{R}_{s}(\mathbf{x})\);
6. Stop when the termination condition is satisfied; otherwise, turn to step 5.  
The flowchart of the fast multi-objective antenna optimization framework based on the MOEAs and improved PSO-RBFNN surrogate model is shown in Figure 7.  
Figure 6: Improved PSO-RBFNN.