## 5 Results and Discussion  
All data for the prediction model include two different data sets, the input data set and the prediction data set, in order to ensure the generalization ability of the network. The input data set is a combination of the orthogonal test results and artificial virtual data in Section 4. The prediction data set comes from the orthogonal test results in Section 2. The Holdout-validation method is used to randomly divide the input data set into three data sets: training data set, validation data set, and test data set according to the proportion of 70%, 15%, and 15%, so as to separate the test data set and the training data set. In addition, the Early stopping strategy is adopted to prevent overfitting. The best accuracy model is selected by Holdout-validation with a certain number of times, and then used to predict the erosion depth. This method is effective and feasible [49]. Table 10 shows the prediction results of BP neural network optimized by different algorithms under different working conditions with the prediction data set. The deviation between the predicted value and the experimental value can be seen in Figure 10. In general, the prediction results of the MAGA-BP model are closer to the points of experimental values. In Figure 11, the fitting results between the prediction results and experimental values of the BP, GA-BP, and MAGA-BP models are shown, and their \(R^{2}\) are 0.9848, 0.9904, and 0.9954, respectively. This shows that all three models have good reliability, and the fitting effect of MAGA-BP is the best. Table 11 shows the five performance indicators of the three prediction models with Errorsum, RMSE, APE\({}_{\text{max}}\), MAPE, and \(R^{2}\). AB\({}_{\text{BP-GABP}}\) represents the percentage change amount of GA-BP model relative to BP model; \(\Delta_{\text{BP-MAGABP}}\) represents the percentage change amount of MAGA-BP model relative to BP model; \(\Delta_{\text{GABP-MAGABP}}\) represents the percentage change amount of MAGA-BP model relative to GA-BP model. It can be seen that the sum of the errors of the BP, GA-BP and MAGA-BP models are 0.3531, 0.2862 and 0.1860, respectively. The Errorsum of the models optimized by GA and MAGA are reduced by 18.93% and 47.31% respectively compared to the BP model. The RMSE optimized by GA and MAGA algorithms reduces the error values from 0.0738 to 0.0530 and 0.0361, respectively, by -28.22% and -51.05%, which has the same order of magnitude in [30; 46; 47; 50]. For APE\({}_{\text{max}}\), the optimization effect of GA is not obvious and it still exceeds 10%, while the value of the model optimized by MAGA is 65.79% lower than it of the GA-BP model, and drops to 4%. The accuracy is significantly improved by MAGA. The MAPE of the BP, GA-BP, and MAGA-BP models are 5.08%, 4.57%, and 2.23%, respectively, of which the value of the MAGA-BP model is the smallest with a decrease of 56%. It can \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Samples No.** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(p\) (MPa) & 4 & 4 & 6 & 6 & 8 & 8 \\ \(\alpha\) (\({}^{\circ}\)) & 60 & 90 & 30 & 45 & 60 & 90 \\ \(t\) (s) & 80 & 60 & 60 & 80 & 40 & 20 \\ Experimental results(cm) & 0.906 & 0.582 & 1.112 & 1.556 & 2.068 & 1.702 \\ BP predicted results(cm) & 0.926 & 0.651 & 1.135 & 1.705 & 2.003 & 1.729 \\ GA-BP predicted results(cm) & 0.854 & 0.648 & 1.097 & 1.609 & 2.047 & 1.782 \\ MAGA-BP predicted results(cm) & 0.922 & 0.592 & 1.128 & 1.493 & 2.106 & 1.745 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of different prediction models. Figure 11: Comparison of the coefficient of determination \(R^{2}\) of different models. Figure 10: Comparison prediction results of different models with experimental values. be seen that the results of all indicators of MAGA-BP are better than the other two models, so the optimization effect of the MAGA algorithm is obvious.