# Response Prediction for Linear and Nonlinear Structures Based on Data-Driven Deep Learning
## 4 Results and Discussion
### Response Prediction for Nonlinear Structure  
Figure 13 shows the probability distribution of the prediction results' normalized errors of the RNN and LSTM models for the nonlinear structure. Within the confidence interval of \([-10\%,10\%]\), the confidence degrees of the three clusters' responses predicted by the RNN model were 90.6%, 82.3%, and 92.6%, respectively, and the confidence degrees of the three clusters responses predicted by the LSTM model were 90.2%, 86.7%, and 95.8%, respectively. Compared with the predicted results for a linear structure, the prediction accuracy of the RNN and LSTM models for nonlinear structure responses was significantly decreased. Table 11 shows the WMAPE_EWMAPE_ and PPE_EE_PEAK_ of the predicted displacement responses \([u_{s6},u_{b},u_{d}]\) of the top layer of the superstructure, linear model \(m_{b}\), and nonlinear model \(m_{d}\). It can be seen that the WMAPE value of the nonlinear system response predicted by the RNN model was small, but the PPE value was large. The results show that, although the RNN model could roughly predict the response trend of the nonlinear structure, to some extent, it was very poor in predicting the peak value of the response. In addition, it could be found that the prediction error of the response to the nonlinear model \(m_{d}\) was significantly higher than that of the top-layer and linear models \(m_{b}\). That is to say, the RNN model could not learn the non-linear features of the response well. Table 12 shows the WMAPE_EWMAPE_ and PPE_PEAK_ of predicted responses \([\tilde{u}_{d},\tilde{u}_{d},u_{b},u_{b},u_{s6},\tilde{u}_{s6}]\). Similar to the prediction results of the RNN model, the prediction responses of the LSTM model also showed the trend of a small WMAPE value and large PPE value. However, compared with the RNN model, the prediction error of the LSTM model was significantly smaller. On the whole, the LSTM model had the worst prediction accuracy for the acceleration response peak value. Figure 13: PDF of prediction results of the RNN and LSTM models for nonlinear systems. (**a**) RNN model; (**b**) LSTM model. In order to further illustrate the prediction accuracy of the RNN and LSTM models for a nonlinear structural response, part of the prediction results and corresponding true values were randomly selected for comparison and analysis, as shown in Figures 14 and 15. It can be seen from Figure 14 that the predicted results of the RNN model were consistent with the overall trend of the true value, but the prediction effect for some details was extremely poor (especially for the reproduction effect of high-frequency characteristics and peak values). The prediction effect of the LSTM model on the nonlinear structural response was significantly better than that of the RNN model (see Figure 15). As shown in Figure 15d,f, the LSTM model could reproduce the details of the response of the nonlinear structure well (although there was some error in the prediction peak value, it was within the allowable range). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Index** & \(\mathbf{u_{66}}\) & \(\mathbf{u_{b}}\) & \(\mathbf{u_{d}}\) \\ \hline \multirow{2}{*}{Cluster 1} & \(E_{WMAPE}\) (\%) & 12.690 & 7.008 & 16.905 \\  & \(E_{PEAK}\) (\%) & 32.366 & 42.877 & 51.623 \\ \hline \multirow{2}{*}{Cluster 2} & \(E_{WMAPE}\) (\%) & 17.913 & 13.923 & 25.308 \\  & \(E_{PEAK}\) (\%) & 21.607 & 58.182 & 66.567 \\ \hline \multirow{2}{*}{Cluster 3} & \(E_{WMAPE}\) (\%) & 10.102 & 9.298 & 20.117 \\  & \(E_{PEAK}\) (\%) & 38.397 & 49.993 & 53.695 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Error values of the RNN model for the nonlinear structure. Figure 14: Comparison of the numerical results and prediction results (RNN model for a nonlinear system, \([u_{66},u_{b}]\)). (**a**) \(u_{66}\) under the action of the seismic wave in cluster 1. (**b**) \(u_{66}\) under the action of the seismic wave in cluster 2. (**c**) \(u_{66}\) under the action of the seismic wave in cluster 3.