## 3 Results and Discussion
### Prediction Effect of Different Surrogate Models  
The training and testing datasets for the surrogate model were derived from simulation results. Therefore, the 137 sets of boundary conditions and EOH results in Section 3.1.2 were employed. They were randomly divided into training and validation sets, with a data volume ratio of 117:20. The BP, DNN, and LSTM models were generated using the same training set and compared to the verification set. The BP model had a 5 \(\times\) 9 \(\times\) 1 structure, that is, five inputs, nine hidden layers, and one output. Three hidden layers were selected to build a DNN model, and the number of neurons in the three hidden layers was 8, 10, and 10, respectively. The number of hidden layers in the first and second layers of the LSTM structure was 5 units and 20 units. The settings of the three prediction models are shown in Table 5. The prediction results of the BP, DNN, and LSTM models are illustrated in Figure 10(a)-c. The horizontal axis represents the results of the three prediction models, while the vertical axis represents the actual results of the simulation model. To visually assess the numerical differences between the horizontal and vertical coordinates, a black dashed line with a slope of 1 was added to observe the distribution of various points. The distribution of the discrete data points on the dashed line can effectively evaluate the performance of each model. The degree of proximity to the dashed line can reflect the accuracy of the model's predictions. As shown in Figure 11, it is evident that all three models performed well in predicting EOH. To further quantify the predictive capabilities of these models, their prediction errors compared to the validation dataset results are listed, and a summary is provided in Table 6.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  &
\begin{tabular}{c} **Maximum** \\ **Relative Error** \\ \end{tabular} & \(\mathbf{R^{2}}\) & **RMSE** & **MAE** \\ \hline BP & 0.030\% & 0.9601 & 0.1130 & 0.0819 \\ DNN & 0.019\% & 0.9734 & 0.0863 & 0.0629 \\ LSTM & 0.014\% & 0.9899 & 0.0511 & 0.0372 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison between the predicted results and the simulation results.  
Figure 10: EOH results obtained through numerical calculation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Activation Function** & **Number of Hidden Layers** \\ \hline BP & \begin{tabular}{c} Tansig for the hidden layer, Purelin for the \\ output layer \\ \end{tabular} & 9 \\ DNN & \
\begin{tabular}{c} ReLU \\ \end{tabular} & 3 \\ LSTM &
\begin{tabular}{c} Sigmoid for the input gate and the forget gate, \\ Tanh for the candidate memory cell \\ \end{tabular} & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Settings of the BP, LSTM, and DNN methods.  
The maximum relative errors of the BP, DNN, and LSTM models were 0.030%, 0.019%, and 0.014%, respectively. The LSTM model performed the best in predicting the EOH value, which could be demonstrated through comparisons of \(R^{2}\), RSME, and MAE. Therefore, this paper employed the LSTM model with the best predictive performance as the surrogate model for rapid EOH prediction.