## 5 Deep Learning Prediction Results and Model Comparison
### Duffing Equation Damped and Forced Vibration Type
In summary, while the LSTM model excels in terms of training time and prediction accuracy, it is prone to fluctuations in training accuracy. Conversely, the LSTM-NN model maintains stable training accuracy but exhibits less accurate predictions, particularly for multiple solutions. Our study primarily focuses on predicting convergence outcomes of the Duffing equation, encompassing cases of +1, -1, and limit cycle oscillations (LCOs). These scenarios are commonly encountered in engineering problems involving nonlinear oscillations. However, it is important to note that our study does not cover cases of chaos or divergence, as they fall outside the scope of this research. The applicability of our models is best suited for scenarios where engineers or researchers need to predict the convergence behavior of systems described by the Duffing equation, which includes situations like aircraft wing vibrations or mechanical component oscillations. Regarding assumptions, we assume that the Duffing equation accurately represents the underlying dynamics of the systems being modeled. Additionally, we assume that the data used for training and validation is representative of real-world scenarios. While our models have shown promising results within the scope of our study, it is important to exercise caution when applying them to systems significantly deviating from the assumptions we have outlined above. \begin{table}
\begin{tabular}{c c c} \hline \hline  & LSTM Max Displ. & LSTM-NN Max Displ. \\ \cline{2-3} MSE (\%) & 0.0484 & 0.0953 \\ RMSE (\%) & 2.1993 & 3.0871 \\ \hline  & LSTM Max Vel. & LSTM-NN Max Vel. \\ \cline{2-3} MSE (\%) & 0.0527 & 0.0960 \\ RMSE (\%) & 2.1812 & 3.0988 \\ \hline  & LSTM Min Displ. & LSTM-NN Min Displ. \\ \cline{2-3} MSE (\%) & 0.0476 & 0.0958 \\ RMSE (\%) & 2.2967 & 3.0948 \\ \hline  & LSTM Min Vel. & LSTM-NN Min Vel. \\ \cline{2-3} MSE (\%) & 0.0381 & 0.0789 \\ RMSE (\%) & 1.9507 & 2.8091 \\ \hline Average & LSTM & LSTM-NN \\ \hline MSE (\%) & 0.0467 & 0.0915 \\ RMSE (\%) & 2.1570 & 3.0225 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: MSE and RMSE error analysis fir the damped system. On considering computational cost, it is important to highlight that our chosen model, the Long Short-Term Memory (LSTM) model, demonstrates notable computational efficiency and low complexity. Firstly, during the training process, we observed that the LSTM model achieved high accuracy within a relatively short amount of time. In fact, as previously mentioned, using a GPU for training proved to be significantly faster compared to a CPU, showcasing the model's computational efficiency. Please see Figures 34 and 35. Secondly, the LSTM architecture is known for its capability to handle sequential data with relatively low computational overhead. This efficiency is due to the architecture's inherent ability to capture and retain important information over time, making it suitable for problems like predicting the convergence outcomes of the Duffing equation. Additionally, our model's straightforward architecture, involving a few hidden layers, contributes to its computational simplicity. This simplicity not only aids in efficient training, but also ensures that the model's inference time, i.e., the time required to make predictions once the model is trained, remains reasonably low. Considering these factors, we can confidently assert that our LSTM model strikes a balance between predictive accuracy and computational efficiency. It offers high accuracy in predicting complex convergence patterns while maintaining efficient training and inference times. Therefore, it effectively addresses the trade-offs involved in model selection, aligning with the study's objectives and requirements.