## 2 Materials and Methods
### Methods
#### 2.3.3 Model Building  
BPNN is a feed-forward neural network composed of an input layer, hidden layers, and an output layer. It iteratively adjusts network weights and thresholds to approximate the target function, thereby achieving the intended model output. The structure of a single  
Figure 2: Research technical route. hidden layer is shown in Figure 3. Feature variables are input into the left side of the input layer, passed through Equation (1) to enter the hidden layer, where they undergo activation through Equations (2) and (3) before being transmitted to the output layer. After passing through another activation function (Equation (4)), the predicted values are generated. Model predictive capabilities are optimized based on the Mean Squared Error (MSE) between predicted and actual values. The BPNN possesses strong self-learning and self-adjustment capabilities, enabling it to exhibit robust fitting and adaptability when dealing with complex data [29,30]. In cases where predicted values deviate from expected values, the error is backpropagated using the gradient descent method to update weights and thresholds. This process influences the final prediction result of BPNN and can be one of the reasons why BPNN is prone to getting stuck in local minima [31]. \[\mathrm{I_{i}}=\sum_{j=1}^{n}\omega_{\mathrm{ij}}\;\times\;\mathrm{x_{j}}\;+ \;\beta_{1}\big{(}0\leq\beta_{1},\;\omega_{\mathrm{ij}}\leq 1\big{)} \tag{2}\]  
\[\mathrm{y_{j}}=\mathrm{f_{1}}\big{(}\mathrm{I_{j}}\big{)} \tag{3}\]  
\[\mathrm{I_{k}}=\sum_{k=1}^{h}\omega_{\mathrm{jk}}\;\times\;\mathrm{y_{j}}\;+ \;\beta_{2}\;\Big{(}0\leq\beta_{2},\;\omega_{\mathrm{jk}}\leq 1\Big{)} \tag{4}\]  
\[\mathrm{y_{t}}=\mathrm{f_{2}}(\mathrm{I_{k}}) \tag{5}\]  
where \(\mathrm{I_{j}}\) and \(\mathrm{I_{k}}\) represent the input values of the hidden layer and output layer, respectively; \(\mathrm{y_{j}}\) and \(\mathrm{y_{t}}\) stand for the output values of the hidden layer and output layer, respectively; \(\omega\) correspond to the weights; \(\beta\) is the thresholds; \(\mathrm{f}\) is the activation functions, with each neuron in the same layer sharing the same activation function; \(\mathrm{n}\) and \(\mathrm{h}\) refer to the number of nodes in the input layer and hidden layer, respectively. To address the issue of BPNN being susceptible to local minima, many scholars have developed intelligent optimization algorithms such as the Genetic Algorithm (GA) [32], Ant Colony Optimization Algorithm (ACO) [33], and the Particle Swarm Optimization Algorithm (PSO) [34]. These optimization algorithms aim to transition the initial weights and thresholds of BPNN from a random state to a well-established state. Compared with  
Figure 3: Structure of BPNN with a hidden layer. other intelligent optimization algorithms, the position of each particle in PSO is updated based on the historical experience of both individual particles and entire group, and the introduction of the inertia weight \(\omega\) of the speed as in Equation (6). The speed of movement of the particle decreases with an increasing number of iterations. This ensures that particles explore a broader range of possibilities during the early optimization stages while accurately converging to the optimal position in the later stages. At this point, the particle's position represents the optimally initialized weights and thresholds for BPNN [35; 36]. This process can be expressed by Equations (7) and (8).