## 3 Improved Sparrow Search Algorithm
### Improved Sparrow Search Algorithm
\begin{table}
\begin{tabular}{c c c c c} \hline
**Type** & **Expression** & **Dimension** & **Range** & **Optimal Value** \\ \hline \multirow{6}{*}{Unimodal function} & F1\((x)=\frac{n}{i}\)\(x_{i}^{2}\) & 30 & \([-100,100]\) & 0 \\ \cline{2-5}  & F2\((x)=\sum\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{n}x_{j}\right)^{2}\) & 30 & \([-100,100]\) & 0 \\ \cline{2-5}  & F3\((x)=\underset{i=1}{\overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{ \overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{ \overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{ \overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{\overset{n}{\overset{i}{ \overset{n}{\overset{i}{\overset{n}{\overset{i}{\overset{n}{\overset{ \oversetFigure 7: _Cont._

### Validation of MISSA-BP Surrogate Model

The structure of the tri-band antenna used in this paper is shown in Figure 8[37]. Figure 9 shows the 3D view of the antenna in the high-frequency structure simulator (HFSS). Since the frequency band of this antenna is within 10 GHz, an FR-4 substrate is used. The dielectric constant of the substrate is 4.4, and the thickness is 1.6 mm. The FR-4 substrate has the characteristics of low cost and high flexibility, which is convenient for making the antenna into a real object later. Table 3 shows the length range of the tri-band antenna sizes. Figure 7: Convergent curve of five algorithms on ten benchmarks functions: (**a**) function F1; (**b**) function F2; (**c**) function F3; (**d**) function F4; (**e**) function F5; (**f**) function F6; (**g**) function F7; (**h**) function F8; (**i**) function F9; (**j**) function F10. Figure 8: Structure diagram of planar tri-band antenna: (**a**) front structure of the antenna; (**b**) reverse structure of the antenna. Figure 5.0 3D view of the planar tri-band antenna in HFSSS: (**a**) front view of the antenna; (**b**) reverse view of the antenna. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **W** & **W1** & **W2** & **W3** & **W4** \\ \hline Range & [22; 26] & [4; 6] & [2.5; 3.5] & [1.5; 2.5] & [0.5; 1.5] \\ \hline
**Parameter** & **L** & **L1** & **L2** & **L3** & **L4** \\ \hline Range & [48; 50] & [18; 20] & [11; 13] & [6; 8] & [8; 10] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Length range of the tri-band antenna sizes (unit: mm). Within the range given in Table 1, using LHS to obtain 100 groups of data. Then the data is simulated by using HFSS 2021 R1, the reflection coefficient is obtained, and 20 sampling frequency points are uniformly selected as a result of set R. 90 of the 100 sets of corresponding data are employed to train the network that we need. The remaining ten sets can be verified for the training effect. The number of input layer nodes of the BP neural network is the same as the antenna structure parameters in Table 1 (i.e., 10). The output layer nodes correspond to the selected 20 sampling frequency points (i.e., 20). To get the desired outcome, the number of hidden layer nodes is determined [15; 25] for five tests, and it is found that selecting 17 nodes is the best. Therefore, the structure adopted by the neural network is 10-17-20, then using MISSA to deal with flaws of networks. According to the parameter settings of the literature and many tests, we determine some parameters of MISSA. The upper limit of the number of iterations is 100, the scale of the initial population is 30, the proportion of the finders is 0.2, the optimization result is substituted into the network structure, and the surrogate model based on MISSA-BP is obtained. To prove relevant indicators of the surrogate model based on MISSA-BP, SVR [5], Kriging [8], PSO-BP, GWO-BP, WOA-BP, SSA-BP, and MISSA-BP surrogate models are constructed for performance comparison. The number of iterations is 200, and the other parameter settings are consistent with Section 5.1. The indicators of evaluation are mean square error (MSE) and mean absolute percentage error (MAPE).