## 2 Establishment of Prediction Model
### Variable-Weight PSO-Optimized BP Neural Network (wPSO-BP) Prediction Model  
The BP neural network (Figure 4) has good nonlinear processing ability and is suitable for predicting the outcomes of the fan. The network is constructed as follows: obtain the predicted values based on the inputs, adjust the node thresholds and connection weights at each layer by calculating the prediction error, and repeat the process until the error is acceptable or the iterative upper limit is reached and then the training is finished. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Boundary Conditions** \\ \hline Blade Inlet Angle & \(\beta_{i}\) & 80\({}^{\circ}\)\(\sim\)115\({}^{\circ}\) \\ \hline Blade Outlet Angle & \(\beta_{o}\) & 129\({}^{\circ}\)\(\sim\)149\({}^{\circ}\) \\ \hline Volute Tongue Radius & \(R_{v,s}\) & 16 mm–25 mm \\ \hline Blade Tip Chamfer Angle & \(\theta\) & 10\({}^{\circ}\)–40\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design parameters and boundary conditions. Figure 3: Beveled blade. Despite the good predictive ability of the BP neural network, it also has problems such as slow convergence speed and difficulty in guaranteeing the global optimization of network parameters, which may not guarantee accuracy of the fan performance prediction. Therefore, this study employs the particle swarm optimization (PSO) algorithm to optimize the BP neural network's initial weights and thresholds, utilizing prediction error as the fitness function to enhance prediction accuracy  
The velocity update formula for particle swarm is as follows:  
\[v_{i}^{k+1}=wv_{i}^{k}+c_{1}r_{1}\left(p_{i}^{k}-x_{i}^{k}\right)+c_{2}r_{2} \left(g_{i}^{k}-x_{i}^{k}\right) \tag{7}\]  
The formula for updating the position of the particle swarm is as follows:  
\[x_{i}^{k+1}=x_{i}^{k}+v_{i}^{k+1} \tag{8}\]  
where \(k\) is the current number of the iteration step, \(w\) is the inertia weight, \(v_{i}^{k}\) is the velocity of particle \(i\) in the \(k\)th iteration, which is a vector when the particle has multiple attributes, \(c_{1}\) and \(c_{2}\) are learning factors used to adjust the maximum step size of particle learning, \(r_{1}\) and \(r_{2}\) are random values within \([0,\ 1]\), and \(x_{i}^{k}\) is the position of particle \(i\) in the \(k\)th iteration. The inertia weight \(w\) controls the degree to which particles preserve their previous velocity during motion updates. A larger \(w\) in the initial phase biases the particle swarm towards global search, while a smaller \(w\) in the later phase helps in local search [20]. In order to better enhance the search ability of PSO, a calculation method for inertia weight decreasing update is proposed:  
\[w=w_{start}\bigg{(}\frac{w_{end}}{w_{start}}\bigg{)}^{\frac{2}{9}\arcsin(\frac {k}{k_{max}})} \tag{9}\]  
where \(w_{start}\) and \(w_{end}\) are the initial and final inertia weights, respectively, and \(k_{max}\) is the upper limit of the number of iterations. Figure 5 illustrates the workflow for constructing a centrifugal fan prediction model based on a variable-weight PSO-optimized BP neural network (wPSO-BP). The model employs a BP neural network incorporating a variable inertia weight strategy, utilizes a rectified linear unit (ReLU) activation function in the hidden layer, applies a linear function in the output layer, and adopts the mean squared error (MSE) as the training objective. The initial learning rate was set to 0.01 and dynamically optimized through the wPSO algorithm. During the data preprocessing stage, all input and output variables were normalized via the Min-Max method, and the dataset was divided into training and testing subsets at a 4:1 ratio to evaluate the model's predictive performance. For the parameter configuration of the wPSO algorithm, the population size was set to 20, the maximum number of iterations was capped at 50, and the inertia weight was linearly reduced from an initial value of 1.0  
Figure 4: The structure of the BP neural network. to 0.4.