## 4 Multi-Objective Optimization
### Hydraulic Performance Prediction Model  
The approximate prediction model uses a small amount of sample data to establish an approximate function expression by interpolation or fitting mathematical methods, so as \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & **Sum of Squares** & **Coefficient** & **Standard** & \multirow{2}{*}{_p-Value_} \\  & **(\%)** & **Assessment** & **Error** & \\ \hline Imported placement angle \(\beta_{1}\) & 4.25 & 0.5951 & 0.4218 & 0.2080 \\ Exit placement angle \(\beta_{2}\) & 15.04 & \(-\)1.12 & 0.4218 & 0.0378 \\ Outlet width \(b_{2}\) & 34.02 & \(-\)1.68 & 0.4218 & 0.0072 \\ Blade wrap angle \(\varphi\) & 95.14 & 2.82 & 0.4218 & 0.0005 \\ Impeller timing position & 1.51 & \(-\)0.3550 & 0.4218 & 0.4322 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Significant analysis of the factors affecting efficiency. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & **Sum of Squares** & **Coefficient** & **Standard** & \multirow{2}{*}{_p-Value_} \\  & **(\%)** & **Assessment** & **Error** & \\ \hline Imported placement angle \(\beta_{1}\) & 0.0010 & \(-\)0.0090 & 0.1078 & 0.9365 \\ Exit placement angle \(\beta_{2}\) & 2.15 & 0.4235 & 0.1078 & 0.0077 \\ Outlet width \(b_{2}\) & 45.48 & 1.95 & 0.1078 & \textless{}0.0001 \\ Blade wrap angle \(\varphi\) & 9.36 & \(-\)0.8830 & 0.1078 & 0.0002 \\ Impeller timing position & 0.0777 & \(-\)0.0805 & 0.1078 & 0.4834 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Significant analysis of influencing factors for the pump head. to carry out follow-up work such as prediction or optimization. The use of approximate predictive model methods can reduce the number of trials and improve the efficiency of optimization. Common approximate models include artificial neural networks, response surface methods, and Kriging models, among which artificial neural networks are widely used in various prediction-type problems with complex relationships because of their ability to adapt to complex non-linear relationships [24]. Therefore, in this paper, an artificial neural network is used to establish a hydraulic performance prediction model for submersible well pumps. As the research progresses, more than forty types of neural network models have been developed, among which BP neural networks and RBF neural networks are widely welcomed by researchers in various fields. BP neural networks are relatively simple in structure, but they can easily be affected by local optimal solutions, and the learning efficiency of the network is fixed, which leads to a large amount of time required to reach convergence criteria. RBF neural networks have a strong generalization capability, good global approximation capability, are relatively unaffected by local optimal solutions, and have an efficient and simple structural form as well as the best approximation performance, so the training speed is faster [25]. Considering the above analysis, the RBF neural network is therefore used to create the prediction model. In this paper, a three-factor, 37-level uniform experimental design was used [26]. Based on ANSYS CFX, 37 groups of models were simulated, and the results were used as training samples. According to the uniform test design table and its use table, the corresponding structural parameters are calculated according to the value range of the outlet placement angle \(\beta_{2}\), the outlet width \(b_{2}\), and the blade wrap angle \(\varphi\), and then the corresponding head and efficiency are calculated based on the corresponding structural parameters. The results are shown in Table 7. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample Serial** & **Exit Placement** & **Outlet** & **Blade Wrap** & **Lift** & **Efficiency** \\
**Number** & **Angle \(\beta_{2}\)** & **Width \(b_{2}\)** & **Angle \(\varphi\)** & **Lift** & **Efficiency** \\ \hline  
5 & 13 & 68 & 27.7594 & 66.0542 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical simulation results of external characteristics.