## 5 Verification Case Study and Discussions
### The Improved PSO-RBFNN Antenna Surrogate Model  
The initial geometry of planar miniaturized multiband antenna is shown in Figure 8 referring to Reference [6]. A rectangular microstrip patch with a circle slot is backed by a FR-4 substrate with a comb-shaped ground. The FR-4 substrate is of thickness 1.6 mm, permittivity 4.4, and loss tangent 0.02. Design parameters are \(\mathbf{x}=\left[d\,l\,l_{1}\,l_{2}\,l_{3}\,l_{4}w\,w_{1}\,w_{2}\,w_{3}\right]^{T}\) and their initial ranges are given in Table 1 (all dimensions in mm).  
Figure 8: The initial geometry of planar miniaturized multiband antenna.  
Figure 7: Flowchart of fast multi-objective antenna optimization framework based on multi-objective evolutionary algorithms (MOEAs) and improved PSO-RBFNN surrogate model.  
When constructing an antenna surrogate model, 200 sample points are randomly initialized as the sample set \(S\) in a given design space \(X\) firstly. These sample points are transmitted to High Frequency Structure Simulator (HFSS) through HFSS-MATLAB-API to obtain the reflection coefficient response set \(Y\). Then the first 180 sample points are used for model training and the remaining 20 are used for testing. The number of input layer neurons is 10, consistent with the number of antenna design parameters to be optimized, and the number of output layer neurons is 15, consistent with the number of frequency points at which reflection coefficients are computed. The number of hidden layer neurons is determined by comparing the errors of RBFNN with different numbers of hidden layer neurons. According to the experience of setting parameter in [35], the parameters of the improved PSO are set as follows: the population size \(M=40\); the maximum iteration number \(MaxIter=1000\); inertia weight \(w=1\); nonlinear time-varying learning factors \(c_{1}\), \(c_{2}\) are adopted in (8) with \(c=2\) and \(\alpha=3\) determined by the experimental tests; the adjusting factor \(\mu=0.5\) is adopted in (9).  
To demonstrate the superiority of the proposed improved PSO-RBFNN model over conventional PSO-RBFNN [24], Figure 9 shows the fitness curve of these two models in the training process. It can be observed that the fitness values of our proposed model are significantly larger than the conventional one, indicating that our model can achieve a much lower training error MSE. Furthermore, Figure 10 compares the scatter plots of the predicted S11 results by Kriging [5], RBFNN [11], PSO-RBFNN [24] and improved PSO-RBFNN relative to the simulations obtained by HFSS. It is clearly observed that the testing points in Figure 10a are more concentrated on the diagonal compared with Figure 10b-d, demonstrating that the proposed model has better prediction ability than Kriging, RBF, and conventional PSO-RBFNN using the same sample points. Besides, the time cost of different surrogate models and HFSS simulations are tabulated in Table 2. The results indicate that the use of various surrogate models greatly reduces the computational time compared to HFSS simulation.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Range** \\ \hline \(d\) & [7; 10] \\ \(l\) & [26; 34] \\ \(l_{1}\) & [11; 14] \\ \(l_{2}\) & [8; 10] \\ \(l_{3}\) & [6; 8] \\ \(l_{4}\) & [10; 14] \\ \(w\) & [17; 23] \\ \(w_{1}\) & [2; 4] \\ \(w_{2}\) & [2; 4] \\ \(w_{3}\) & [0.5, 1.5] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial range of design parameters (units: mm).  
Figure 10: Comparison of actual S\({}_{11}\) values when using **(a)** Improved PSO-RBFNN, **(b)** PSO-RBFNN, **(c)** RBFNN, **(d)** Kriging surrogate models.  
Figure 9: Fitness curve of conventional PSO-RBFNN and improved PSO-RBFNN model in the training process.  
In summary, our proposed PSO-RBFNN model can be used for performance prediction efficiently instead of EM simulation software and achieve a fast multi-objective optimization with the help of MOEAs for a predefined antenna geometry.