## 3 Methods
### Parameter Debugging  
(1) Number of nodes per layer  
Increasing the number of neuron nodes generally enhances the training effectiveness of BP neural networks, effectively reducing the overall error in predictions (Figure 4). In practical applications, the quantity of nodes is typically iteratively adjusted to optimize network performance. In this case, after conducting 250 trials to refine the model, the optimal network configuration was established as having 50 nodes in the input layer, 90 nodes in the hidden layer, and 90 nodes in the output layer. The specific configuration demonstrated robust performance, achieving an R\({}^{2}\) score of 0.7719 and a remarkably low error rate of 0.0015, indicating a high degree of predictive accuracy and model reliability.  
(2) Learning rate and momentum  
The learning rate is a critical parameter in neural networks that directly influences the magnitude of weight adjustments during training. Both excessively high and low learning rates can negatively impact the performance of a backpropagation (BP) neural network, leading to either erratic or insufficient updates to the model's weights. Similarly, the momentum parameter plays a pivotal role in determining the consistency of the gradient's direction across updates, enhancing the training process when consistent and moderating updates otherwise. In our experiments, we conducted multiple tests on the BP neural network, varying the momentum between 0.1 to 0.9 and adjusting learning rates from 0.01 to 0.1. Through these tests, we identified that the network achieved an optimal performance with a momentum of 0.5 and a learning rate of 0.04, resulting in an R\({}^{2}\) score of 0.7688 (Figure 5). This combination of learning rate and momentum was shown to be the most effective in achieving a balance between the rapid convergence and stability of the learning process. The learning rate, which is a crucial neural network parameter, influences the weight changes' magnitude. Both overly high and low learning rates adversely affect the BP neural network. The momentum determines the consistency of the gradient direction, boosting updates if consistent and reducing them otherwise. Multiple tests were conducted on our BP neural network, with the momentum varying from 0.1 to 0.9 and learning rates from 0.01 to 0.1. The optimal result with an R\({}^{2}\) score of 0.7688 was achieved at a momentum of 0.5 and a learning rate of 0.04.  
(3) Excitation function  
In BP neural networks, excitation functions are utilized for non-linear processing. While the output layer typically uses sigmoid as the default, this study tested various combinations of excitation functions. After nine trials, the optimal configuration was identified as tanh for the input layer and sigmoid for the output layer. This combination yielded an R\({}^{2}\) score of 0.6952 and an error of 0.0018 (Table 1).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Input Layer** & **Hidden Layer** & **Error** & **R\({}^{2}\)** \\ \hline Sigmoid & Sigmoid & 0.006195 & 0.0238 \\ Sigmoid & Rule & 0.003329 & 0.4171 \\ Sigmoid & Tanh & 0.002153 & 0.6487 \\ Rule & Sigmoid & 0.002409 & 0.5743 \\ Rule & Rule & 0.002267 & 0.5972 \\ Rule & Tanh & 0.001976 & 0.6877 \\ Tanh & Sigmoid & 0.003035 & 0.5906 \\ Tanh & Rule & 0.001943 & 0.6917 \\ Tanh & Tanh & 0.001840 & 0.6952 \\ \hline \hline \end{tabular}
\end{table}
Table 1: R\({}^{2}\) score and error of excitation function.  
Figure 4: R\({}^{2}\) score and error of neurons in each layer.