## 3 Structural Optimization of PIPR's End Face
### Pressure Gradient Model Based on BP Neural Network  
In order to analyze the influence of four optimization variables on the average pressure gradient, the significance test is performed, as shown in Table 5. The \(p\) value of less than 0.05 indicates that the variables are significant. It can be seen that the influence of \(h\) and \(S_{p}\) is greater than that of the other two variables. The significance of the four variables is ranked as \(S_{p}>h>r>s\). The interactive influence of optimization variables is shown in Figure 19. It can be seen that the curved surfaces are a downward depression. The influencing trend of the four optimization variables is different, and the influence of \(S_{p}\) is the most obvious. With the increase in \(S_{p}\), the average pressure gradient first decreases and then increases. When \(S_{p}\) is small, with the increase in \(h\), the average pressure gradient gradually increases; when \(S_{p}\) is large, with the increase in \(h\), the average pressure gradient first increases and then decreases. For \(s\), when \(S_{p}\) is small, with the increase in \(s\), the average pressure gradient gradually increases, but the trend is contrary when \(S_{p}\) is large. For \(r\), when \(S_{p}\) is small, with the increase in \(r\), the average pressure gradient gradually decreases, but the trend is contrary when \(S_{p}\) is large. BP neural network is a multi-layer feed-forward network trained by the error inverse propagation algorithm. It uses the gradient descent method to continuously adjust the weights and thresholds so that the error is minimized. Then, the nonlinear mapping relationship between the input data and output data is established. Due to the advantages \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number** & \(s\)**/cm** & \(h\)**/cm** & \(r\)**/cm** & \(S_{p}\) & \(\frac{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\cdot}}}}}}}}}}}}} \)/**P/m** \\ \hline
1 & 22.45 & 10.73 & 4.99 & 0.66 & 31,254.11 \\
2 & 14.60 & 11.93 & 3.76 & 0.10 & 28,827.85 \\
3 & 16.97 & 5.22 & 7.29 & 0.78 & 30,020.79 \\
4 & 18.40 & 8.04 & 6.28 & 0.99 & 32,902.59 \\
5 & 20.27 & 9.39 & 5.58 & 0.47 & 29,739.70 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The simulation results of different parameters of the end face. Figure 19: The interactive influence of optimization variables. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & \(F\)**Value** & \(p\)**Value** \\  & & **Prob \textgreater{} \(F\)** \\ \hline \(s\) & 1.59 & 0.2268 \\ \(h\) & 5.38 & 0.0348 \\ \(r\) & 2.17 & 0.1615 \\ \(S_{p}\) & 11.84 & 0.0036 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The results of the significance test. of nonlinear fitting of the BP neural network, the BP model for PIPR's end face parameters and average pressure gradient is constructed, as shown in Figure 20. The BP model includes one input layer, one hidden layer, and one output layer. The neuro number of the hidden layer is 5. The maximum number of training epochs and learning rate of the gradient descent algorithm is set to 100 and 0.1, respectively. Eighty percent of experimental samples are selected as a training set, and twenty percent of experimental samples are selected as a test set. The prediction results of the average pressure gradient based on the BP neural network are shown in Figure 21. It can be noticed that the prediction results are roughly all distributed around a curve with a slope of 1, which indicates that the BP model has a good prediction performance. To verify the advantages of the established BP neural network model, the support vector machine (SVM), generalized regression neural network (GRNN), random forest (RF), and response surface method are used for comparison. Three evaluation metrics are selected to measure the regression effect, namely the mean average error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R\({}^{2}\)).