## 3 Results and Discussion
### Determination of the Optimal BPNN Performance  
The network structure parameters were determined, and then the BPNN was trained. After multiple-network training, the networks with excellent performance were saved, and the network performance data were recorded. All network performance data were compared, and the optimal network was selected for subsequent research.  
Table 18 presents the performance data of the optimal network. The R\({}^{2}\) values of two outputs for the training set are both above 0.96, demonstrating the high fitting degree of the network to the data in the training set. Moreover, the strong generalization ability of the network is manifested by the fact that the R\({}^{2}\) values of two outputs for the test set are both above 0.92. The values of the error indices are basically within 2%, remaining at a relatively low level, indicating that the network has a small predictive error.  
Use the sim function in MATLAB R2022a to obtain the predicted values of the outputs (_MI_ and _RI_) corresponding to the inputs of the training and test sets, and then compare them with the experimental values (as shown in Tables 6-11), as shown in Figures 11 and 12. In the training sample part, the predicted curve basically coincides with the experimental curve, and the maximum absolute errors of the _MI_ and _RI_ are 6.56% and 4.42%, respectively; in the test sample part, except for a few individual samples, the predicted curve fits the experimental curve well, and the maximum absolute errors of the _MI_ and _RI_ are 5.01% and  
Figure 10: Comparison of optimal network performance under each activation function combination (R\({}^{2}\), RMSE, and MAE denote the coefficient of determination, root mean square error, and mean absolute error between experimental and BPNN-predicted values, respectively; \({}^{-}\) denotes average value of network performance indices for missing index and reseeding index; TR and TS denote training set and test set, respectively).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Network**} & \multirow{2}{*}{**Output**} & \multirow{2}{*}{**Dataset**} & \multicolumn{4}{c}{**Performance Indices**} \\ \cline{3-5}  & & & **RMSE** & **MAE** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{BPNN (7-8-2)} & Missing & Training set & 1.4883 & 1.0937 & 0.9627 \\  & index & Test set & 1.8807 & 1.3440 & 0.9287 \\ \cline{1-1}  & Reseeding & Training set & 1.2064 & 0.8436 & 0.9753 \\ \cline{1-1}  & index & Test set & 2.0178 & 1.7559 & 0.9497 \\ \hline \hline \end{tabular}
\end{table}
Table 18: Performance of the optimal performance network (R\({}^{2}\), RMSE, and MAE denote the coefficient of determination, root mean square error, and mean absolute error between experimental and BPNN-predicted values, respectively).  
3.80%, respectively. The comparison between the experimental and predicted values of the _MI_ and _RI_ also demonstrates the excellent fitting performance and strong generalization ability of the selected network.  
The weights and biases of the optimal performance BPNN were extracted for the further combination of the BPNN predictive model and MOPSO algorithm, as shown in Table 19.