## 3 Prediction Method
### Multi-Neural Network Ensemble Model  
The multi-neural network integration model has multiple BPNNs, with multiple fully connected layers as hidden layers; each input neuron is connected to each output neuron. A hyperparameter optimization process was performed to construct the optimal integration structure. The hyperparameters considered include the number of hidden layers, the number of neurons in each hidden layer, and the type of activation function. The training data set was used to train each artificial neural network during the network training process. The validation set is used to detect and prevent overfitting. BPNN systematically solves the problem of learning the connection rights of hidden layers in multilayer networks by the gradient descent method, where the input variables are fed from the input layer and processed by the hidden layer to reach the output layer. The error between the output and actual values is calculated, and the error information is fed back to the network through backpropagation. We use the gradient descent method to repeatedly adjust the connection weights and thresholds between layers until the global error of the network is minimized so that the network output approximates the actual output. The path loss prediction results for a single BPNN model can be expressed as follows:  
\[PL_{BPNN}=F_{BPNN}\big{(}R_{s},C_{s},C,d,\Delta_{x},\Delta_{y},h_{R}|w,b, \sigma(\cdot)\big{)} \tag{6}\]  
where \(R_{s}\), \(C_{s}\), \(C\), \(d\), \(\Delta_{x}\), \(\Delta_{y}\) and \(h_{R}\) denote the row number of the carriage seat, the column number of the carriage seat, the carriage number, the Euclidean distance of the transceiver antenna (m), the relative distance of the X-axis of the transceiver antenna (m), the relative distance of the Y-axis of the transceiver antenna (m), and the height of the receive antenna, respectively. \(w\), \(b\) and \(\sigma(\cdot)\) denote the weight matrix, bias matrix, and activation function. The model consists of an input layer, two hidden layers, and an output layer, with the output denoted as:  
\[PL_{BPNN}=\sigma_{0}\left(\sum_{k=1}^{N_{2}}w_{1,k}^{(3)}\times\sigma_{H}\left( \sum_{n=1}^{N_{1}}w_{k,n}^{(2)}\times\sigma_{H}\left(\sum_{i=1}^{7}w_{n,i}^{(1 )}\times x_{i}+b_{n}^{(1)}\right)+b_{k}^{(2)}\right)+b^{(3)}\right) \tag{7}\]  
where \(N_{1}\) is the number of contains in the first hidden layer, with weights and biases from the input features to the neurons in the first hidden layer denoted by \(w_{n,i}^{(1)}\) and \(b_{n}^{(1)}\), respectively, with the activation function as \(\sigma_{H}\). The second hidden layer contains  
Figure 7: Field strength prediction method for HSR carriages. neurons, with weights and biases from the first to the second hidden layer denoted by \(w_{k,n}^{(2)}\) and \(b_{k}^{(2)}\), and from the second hidden layer to the output layer denoted by \(w_{1,k}^{(3)}\) and \(b^{(3)}\). The activation function of the output layer \(\sigma_{O}\) is the identity function. By training networks with different hyperparameters, a large number of BPNN models are obtained. In the prediction phase, to more accurately integrate the predictions of multiple models with different hyperparameters, we assign different weights to the optimal BPNN models, thus achieving a weighted ensemble. Let the predicted path loss of the \(r\)-th model be \(PL_{r}\), and its corresponding weight be \(w_{r}\). The final predicted path loss of the ensemble model is then:  
\[PL_{ANNs}=\sum\limits_{r=1}^{R}w_{r}\times PL_{r} \tag{8}\]  
The above method shows that the ensemble multi-neural network model outperforms single models and ensemble methods like Boosting and Bagging in diversity, generalization, flexibility, and stability. It captures diverse features and enhances robustness by leveraging different network architectures and training subsets. This approach better integrates multiple influencing factors in high-speed rail field strength prediction, where Boosting and Bagging struggle with complex dependencies.