# Deep Learning-Based FSS Spectral Characterization and Cross-Band Migration
## 3 Cross-Band Transfer Learning for FSS
The formulas are as follows [30]:  
\[MAE=\frac{1}{N}\sum\limits_{i=1}^{N}\left|T_{i}^{\prime}-T_{i}\right| \tag{5}\]  
\[RMSE=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}\left(T_{i}^{\prime}-T_{i}\right)^ {2}} \tag{6}\]  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{N}\left(T_{i}^{\prime}-T_{i}\right)^{2}}{ \sum\limits_{i=1}^{N}\left(T_{i}^{\prime}-\overline{T_{i}}\right)^{2}} \tag{7}\]  
where \(T_{i}^{\prime}\) is the CNN prediction result, \(T_{i}\) is the COMSOL simulation result, \(\overline{T_{i}}\) is the average value of the simulation result, and \(N\) is the number of frequency points. Figure 9: Illustration of Convolutional Neural Network Training Process. (**a**) Loss value variation graph; (**b**) RMSE variation graph. The training process is shown in \[MSE=\frac{1}{2}\sum_{i=1}^{N}\left(T_{i}^{\prime}-T_{i}\right)^{2} \tag{3}\]0 It can be observed that due to the reduced amount of data, the training results quickly reached stability. After training, the objective evaluation metrics were as follows: MAE was 0.0719, RMSE was 0.1267, and R-squared (R\({}^{2}\)) was 0.8145. Compared to the objective evaluation metrics from training from scratch (MAE of 0.0592, RMSE of 0.1005, and R\({}^{2}\) of 0.6692), both MAE and RMSE increased, indicating that the errors had grown and the performance of the neural network had worsened. However, R\({}^{2}\) had increased, indicating improved fit, and thus the neural network's performance appeared better, showing a contradiction. This is because the data used in millimeter-wave transfer learning were a subset of terahertz-wave data, which enhanced the prediction results. In a comprehensive analysis, the performance of the neural network trained with transfer learning should be weaker than that of the network trained from scratch, but transfer learning only used 23.1% of the former's data, and both methods showed comparable performance. This makes transfer learning an important and meaningful neural network training approach. After the training of the transfer learning neural network was completed, some of the results were randomly selected for comparison. The comparison results are shown in \[MSE=\frac{1}{2}\sum_{i=1}^{N}\left(T_{i}^{\prime}-T_{i}\right)^{2} \tag{3}\]1 It was found that both the training set and the test set had a high degree of fit, which is comparable to the effect of the neural network trained from scratch by the THz CNN. After completing the training of the transfer learning neural network, the convolutional neural network and the genetic algorithm can be jointly used to complete the design of various types of typical FSSs in microwave bands. It is worth mentioning that this paper only takes microwave bands as an example to show that this method of transfer learning is feasible in cross-band FSS design. Advantages of concise and efficient deep learning can be further brought out due to the reduction in the amount of original data from 26,000 to 6000 groups, which is a reduction of 76.9%. The method can be applied to other bands where necessary through the same steps to expand the method from solving a specific problem to solving a certain class of problems, greatly expanding the application range of the method. \[MSE=\frac{1}{2}\sum_{i=1}^{N}\left(T_{i}^{\prime}-T_{i}\right)^{2} \tag{3}\]2 Transfer Learning Training Process. (**a**) Loss value variation graph; (**b**) RMSE variation graph. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**CNN** & **MAE** & **RMSE** & **R\({}^{2}\)** \\ \hline THz & 0.0592 & 0.1005 & 0.6692 \\ MM-wave & 0.0719 & 0.1267 & 0.8145 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of CNN prediction results for THz and millimeter bands.