## 3 Improved Sparrow Search Algorithm
### Improved Sparrow Search Algorithm
\[MSE=\frac{1}{20}\sum\limits_{n=1}^{20}\big{|}R(n)-\overline{R}(n)\big{|}^{2} \tag{13}\]

\[MAPE=\frac{\sum\limits_{n=1}^{20}\big{|}\frac{R(n)-\overline{R}(n)}{R(n)}\Big{|} }{20}\times 100\% \tag{14}\]

where \(R(n)\) is the actual data of the electromagnetic software and \(\overline{R}(n)\) is the simulation data obtained from the MISSA-BP model. Each optimized surrogate model performs ten independent predictions on the test set, and Table 4 shows the average accuracy of prediction and time-consuming. The average training fitness iteration curves of PSO-BP, GWO-BP, WOA-BP, SSA-BP, and MISSA-BP surrogate models are shown in Figure 10.

From the data in Table 4, we can see that both MES and MAPE of the MISSA-BP surrogate model are obviously smaller than the other models, the accuracy is higher, and the time is shorter. Figure 10 shows that the MISSA-BP has the best fitness compared with other models in the same iteration times, and with the increase of iteration times, the performance of MISSA-BP becomes better and better. Therefore, the MISSA-BP model can substitute for the high-fidelity electromagnetic software and simultaneously design the antenna with the MOPSO algorithm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Surrogate Model** & **MSE** & **MAPE/ (\%)** & **Time-Consuming (s)** \\ \hline SVR [5] & 3.86 & 19.69 & 15.56 \\ Kriging [8] & 3.78 & 16.76 & 14.33 \\ BP & 3.70 & 15.68 & 10.12 \\ PSO-BP & 2.74 & 11.35 & 9.92 \\ GWO-BP & 3.05 & 11.87 & 9.61 \\ WOA-BP & 3.42 & 12.02 & 9.57 \\ SSA-BP & 2.05 & 10.15 & 9.32 \\ MISSA-BP & 0.87 & 7.14 & 7.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prediction accuracy of each optimized agent model. ### Design of Multi-Objective Antenna

Using the MISSA-BP model constructed in part 5.2 united with MOPSO, the antenna is designed with multiple objectives. The design optimization objective of the antenna: (1) Keep the antenna as small as possible, i.e., \(F_{1}=W\times L\). (2) Make the center frequency point of the antenna in the following range, 1.97-2.29 Ghz, 4.88-5.46 Ghz, and 7.18-7.54 Ghz works in S and C bands and can be applied to WLAN wireless network and high-frequency communication. The equations are as follows:

\[F_{2}=\frac{\sum\limits_{n=1}^{20}f(x_{n})}{20} \tag{15}\]

\[\text{where }f(x_{n})= \begin{cases}S_{11}(x_{n})&S_{11}(x_{n})<-10\\ -10&S_{11}(x_{n})>-10\end{cases} \]

where \(x_{n}\) is the \(n\)th sampling frequency point; \(S_{11}(x_{n})\) is the reflection coefficient \(S_{11}\) of the corresponding frequency point. After multiple tests, the MOPSO parameters are set as below: particle swarm scale is 200, the dimension of the particle is 10, the upper limit of velocity is 1, the number of iterations is 300, and the weight parameter is reduced regular from 1 to 0.5, and the learning factor is 2.05. The MISSA-BP surrogate model and MOPSO start to iterate together to optimize objectives F1 and F2. The range of optimized antenna size parameters is given in Table 3. When MOPSO reaches the set number of iterations, the procedure ends, and the obtained Pareto Front is shown in Figure 11. In Figure 11, each point represents an antenna model. The antenna sizes can be obtained from the output data, which are within the specified range. The procedure is run repeatedly five times, each time selecting one from the obtained set of solutions. Table 5 shows the detailed sizes of the five antennas. It can be seen from Table 5 that all ten dimensions are within the range specified in Table 3 and meet the design requirements. Figure 10: Iteration curve of the average fitness of the surrogate model. The reflection coefficient curves of the five groups of antennas simulated in HFSS software look like Figure 12. Furthermore, five groups of antenna models are simulated with the HFSS software for the 3D polar plot, and the results are shown in Figure 13. The 3D polar plot of the antenna is an important graph to measure the performance of the antenna, from which we can observe the parameters of the antenna, such as radiation direction, gain, etc.