## 4 Optimization and Analysis of Multi-Blade Centrifugal Fan
### Prediction Model Results and Analysis  
To establish the mapping between design variables and optimization objectives, 80 sample points are generated using the Latin hypercube sampling (LHS) method [29]. The use of LHS ensures uniform coverage of the multidimensional parameter space with a limited number of samples, thereby preventing local clustering while enhancing predictive accuracy and reducing computational costs [30]. These samples, including representative cases listed in Table 4, constitute the training and testing datasets for the prediction model. The performance metrics (e.g., total pressure, efficiency, and sound pressure level) for these samples are obtained through validated simulation techniques. Moreover, an automated batch processing workflow is implemented to handle the computational workload, which substantially improves computational efficiency.  
The comparison results of wPSO-BP and BP are presented in Figure 12. Compared to the BP prediction model, the value obtained by the wPSO-BP prediction model is closer to the calculated value.  
Figure 11: Noise spectrum of simulation and experiment.  
Figure 10: Experimental research on noise of the prototype fan. (**a**) Semi-anechoic room. (**b**) Microphone position.  
To further compare the two prediction models, quantitative analysis is conducted. The root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R\({}^{2}\)) are calculated according to Equation (16)-(18), respectively.  
\[RMSE=\sqrt{\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y_{i}})^{2}} \tag{16}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{2}{c}{**Design Variables**} & \multicolumn{4}{c}{**Optimization Objectives**} \\ \hline \(\beta_{i}/^{\circ}\) & \(\beta_{o}/^{\circ}\) & \(R_{v,s}/\)mm & \(\theta/^{\circ}\) & \(P_{total}/\)Pa & \(\eta/\%\) & \(L_{p}/\)dB \\ \hline
99.72 & 141.49 & 23.65 & 13.25 & 1237.46 & 56.48 & 56.39 \\ \hline
100.57 & 143.81 & 19.53 & 28.7 & 1268.53 & 57.4 & 56.73 \\ \hline
102.94 & 133.66 & 17.93 & 22.98 & 1175.38 & 55.18 & 55.85 \\ \hline
102.28 & 141.84 & 19.86 & 14.79 & 1226.4 & 55.47 & 55.63 \\ \hline
82.98 & 132.66 & 24.39 & 33.03 & 1183.22 & 57.77 & 56.77 \\ \hline
104.52 & 129.22 & 19.25 & 14.45 & 1074.04 & 52.91 & 53.87 \\ \hline
93.33 & 135.77 & 19.04 & 38.95 & 1168.31 & 56.46 & 54.11 \\ \hline
93.83 & 131.69 & 19.39 & 34.81 & 1114.82 & 52.79 & 52.26 \\ \hline
84.12 & 135.42 & 20.68 & 16.18 & 1223.67 & 59.41 & 56.83 \\ \hline
93.19 & 146.36 & 23.73 & 34.01 & 1314.61 & 59.05 & 57.64 \\ \hline \end{tabular}
\end{table}
Table 4: Information of some sample points and calculations.  
Figure 12: Comparison between predicted and calculated values. (**a**) Total pressure. (**b**) Efficiency. (**c**) Sound pressure level.  
\[MAE=\frac{1}{m}\sum_{i=1}^{m}\left|(y_{i}-\hat{y}_{i})\right| \tag{17}\]  
\[R^{2}=1-\frac{\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{m}(y_{i}- \overline{y})^{2}} \tag{18}\]  
where \(m\) is the number of samples, \(\hat{y}_{i}\) is the true value corresponding to sample \(i\), and \(\overline{y}\) is the average of true values.  
Table 5 shows the evaluation indexes of the two models; it can be seen that the wPSO-BP prediction model can effectively reflect the link between the design variables and the optimization objectives, and its coefficient of determination is greater than 0.9, which can be used for subsequent optimization.