## 4 Multi-Objective Optimization
### Multi-Objective Particle Swarm Optimization Algorithm Optimization  
The particle swarm optimization continuously adjusts the particles' own velocity towards the extreme value during the computational solution process without setting empirical parameters as in the genetic algorithm, and the better particle positions in the swarm algorithm are recorded. Compared with genetic algorithms, particle swarm optimization can converge faster in most cases [27]. Therefore, particle swarm optimization is chosen to solve the neural network prediction fitness function model in this paper. The particle swarm optimization has two core formulas: the velocity formula and the position formula, as shown in the following equation:  
\[V_{i}^{d}=\omega v_{i}^{d}+c_{1}r_{1}(p_{i}^{d}x_{i}^{d})+c_{2}r_{2}(p_{g}^{d}- x_{i}^{d}) \tag{14}\]  
\[X_{i}^{d}=x_{i}^{d}+v_{i}^{d} \tag{15}\]  
where \(X_{i}=(x_{i}^{1},x_{i}^{2},\ldots,x_{i}^{D})\) is a solution in D-dimensional space; \(P_{i}=(p_{i}^{1},p_{i}^{2},\ldots,p_{i}^{D})\) is the position nearest to the optimum among all positions; \(P_{g}=(p_{g}^{1},p_{g}^{2},\ldots,p_{g}^{D})\) is the position nearest to the optimum among all positions through which all particles pass; \(V_{i}=(v_{i}^{1},v_{i}^{2},\ldots,v_{i}^{D})\) is the velocity of the particle; \(\omega\) is the inertia weight; \(c_{1}\), \(c_{2}\) is the learning factor; and \(r_{1}\), \(r_{2}\) is any random number between 0 and 1. Figure 8: External characteristic prediction error. (**a**) Lift; (**b**) efficiency. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Serial** \\ **Number** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Exit** \\ **Placement** \\ **Angle** \(\beta_{2}\) \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Outlet** \\ **Width** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Blade** \\ **Wrap** \\ **Angle** \(\varphi\) \\ \end{tabular} } & \multicolumn{2}{c}{**Lift**} & \multicolumn{2}{c}{**Efficiency**} & \multirow{2}{*}{**Error**} \\ \cline{3-3} \cline{8-10}  & & & **Calculated** & & & **Calculated** & & **Calculated** & **Predicted** \\  & & & **Angle** \(\varphi\) & & **Value** & & & **Value** & \\ \hline
1 & 17 & 10 & 102 & 23.3586 & 23.0377 & \(-\)1.38\% & 76.9654 & 77.7294 & 0.99\% \\
2 & 27.5 & 12.2 & 72 & 26.6137 & 26.8979 & 1.07\% & 66.8634 & 67.6188 & 1.13\% \\
3 & 23 & 11 & 98 & 24.7361 & 24.601 & \(-\)0.55\% & 74.3373 & 75.3969 & 1.43\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Error analysis of numerical calculation results and prediction model results. In this paper, the multi-objective particle swarm optimization is used, based on the MATLAB software platform for programming simulation, and the sample results of the RBF neural network training are used as the fitness model for this optimization, with parameters set as follows: the number of particles is 100, the number of particle reserves is 100, the inertia weight is 0.7298, and both learning factors are 1.49445. After 500 iteration steps, the results are shown in Figure 9. It can be seen from the figure that the horizontal coordinate efficiency and the vertical coordinate lift have a mutually exclusive relationship, and the 100 Pareto optimal solutions obtained constitute a relatively smooth curve. The optimal solutions are also more evenly distributed, so the solution can be considered correct and reliable. The edge points on both sides of the Pareto front are defined as the head optimum and the efficiency optimum resulting from the solution. The initial model is simulated by the CFD method. The optimal head individual is the model at the optimal head point predicted by the MOPSO algorithm. The optimal efficiency individual is the model at the optimum efficiency point as predicted by the MOPSO algorithm. Table 9 shows the comparison of the hydraulic performance of the submersible well pump before and after optimization.