# Determining Rock Joint Peak Shear Strength Based on GA-BP Neural Network Method
## 3 Peak Shear Strength Prediction Model
### Forecasting Results  
After determining the structure, the GA-BP neural network is trained under the optimization of a genetic algorithm. The \(R^{2}\), _MAE_, _MBE_, and _RMSE_ of the training results of each neural network model are shown in Table 5. At the same time, Table 5 also shows the prediction results of the BP neural network without genetic algorithm optimization. It can be seen that the neural network model optimized by genetic algorithm is better. It can be seen from Table 5 that the \(R^{2}\) of each GA-BP neural network model is above 0.93, and the \(R^{2}\) of the six-factor training set reaches 0.96, indicating that the trained model with more input factors has a better prediction accuracy. Using Tables 4 and 5 for comparison, the indicators of each GA-BP neural network model are basically better than the JRC-JCS model.  
In addition to observing _MAE_, _MBE_, _RMSE_, the prediction accuracy of the GA-BP neural network can be observed by linear fitting of real value and predicted value. As shown in Figure 8, the linear fitting plots of real value and predicted value of training set and test set in each model are drawn. It can be seen from Figure 8 that the fitting lines of the real value and the predicted value are very close to y = x, and \(R^{2}\) is above 0.9. As the number of input factors increases, the \(R^{2}\) of the training set also increases.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & Factors & \begin{tabular}{c} GA-BP \\ four-factor \\ model \\ \end{tabular} & \begin{tabular}{c} GA-BP \\ five-factor-1 \\ model \\ \end{tabular} & \begin{tabular}{c} GA-BP \\ five-factor-2 \\ model \\ \end{tabular} & \begin{tabular}{c} GA-BP \\ five-factor-2 \\ model \\ \end{tabular} \\ \hline \multirow{4}{*}{Training set} & \(R^{2}\) & 0.94 & 0.95 & 0.95 & 0.96 \\  & _MAE_ & 0.21 & 0.20 & 0.21 & 0.17 \\  & _MBE_ & 0.04 & 0.01 & 0.00 & 0.01 \\  & _RMSE_ & 0.31 & 0.28 & 0.28 & 0.24 \\ \hline \multirow{4}{*}{Test set} & \(R^{2}\) & 0.93 & 0.93 & 0.95 & 0.93 \\  & _MAE_ & 0.22 & 0.23 & 0.17 & 0.23 \\  & _MBE_ & 0.14 & 0.04 & \(-\)0.02 & 0.06 \\  & _RMSE_ & 0.31 & 0.33 & 0.24 & 0.37 \\ \hline \multirow{4}{*}{Training set} & Factors & \begin{tabular}{c} BP four-factor \\ model \\ \end{tabular} & \begin{tabular}{c} BP five-factor-1 \\ model \\ \end{tabular} & \
\begin{tabular}{c} BP five-factor-2 \\ model \\ \end{tabular} &
\begin{tabular}{c} BP six-factor \\ model \\ \end{tabular} \\ \hline \multirow{4}{*}{Training set} & \(R^{2}\) & 0.92 & 0.92 & 0.94 & 0.94 \\  & _MAE_ & 0.24 & 0.26 & 0.24 & 0.22 \\  & _MBE_ & \(-\)0.01 & \(-\)0.09 & 0.03 & \(-\)0.01 \\  & _RMSE_ & 0.33 & 0.34 & 0.31 & 0.27 \\ \hline \multirow{4}{*}{Test set} & \(R^{2}\) & 0.90 & 0.92 & 0.93 & 0.94 \\  & _MAE_ & 0.36 & 0.25 & 0.25 & 0.29 \\ \cline{1-1}  & _MBE_ & \(-\)0.04 & 0.34 & 0.11 & \(-\)0.05 \\ \cline{1-1}  & _RMSE_ & 0.47 & 0.33 & 0.32 & 0.36 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Training results of neural network models.  
Figure 8: Linear fitting of real value and predicted value: (**a**) four-factor model training set; (**b**) four-factor model test set; (**c**) five-factor-1 model training set; (**d**) five-factor-1 model test set; (**e**) five-factor-2 model training set; (**f**) five-factor-2 model test set; (**g**) six-factor model training set; (**h**) six-factor model test set.