## 3 The Proposed Method and Simulation Verification
### The Proposed Method  
Initially, the features of the acceleration response are extracted by the CNN, as shown in Section 2.2. Additionally, the Mish function [49] is used in the convolution layer because of its remarkable accuracy and generalization ability. \[\tanh(x_{x})=\frac{e^{x_{x}}-e^{-x_{x}}}{e^{x_{x}}+e^{-x_{x}}}, \tag{40}\]  
\[\text{M}(x_{x})=x_{x}\times\tanh[\ln(1+e^{x_{x}})], \tag{41}\]  
\[\textbf{Y}_{P(i,j,k)}=\max(\text{M}(b_{j}+\sum_{k=0}^{C_{\text{in}}-1}w_{jk}* \text{R}_{ik})). \tag{42}\]  
Figure 1: The structural diagram of the simulation example. Figure 2: The process of dataset construction. Here, the loss function of the whole network structure is the mean square error (MSE). \[\text{MSE}=\frac{1}{n}\sum_{i=1}^{n}{(y_{i}-\hat{y}_{i})^{2}}, \tag{43}\]  
where \(n\) denotes the total number, \(y_{i}\) and \(\hat{y}_{i}\) are the actual and the predicted output value, respectively. In the second step, GRU is used to learn the features extracted by CNN. \[\mathbf{Y}_{G}=\text{G}(\mathbf{Y}_{P}), \tag{44}\]  
where G (\(\cdot\)) denotes the GRU networks calculation, and it is detailed in Section 2.3. \(\mathbf{Y}_{P}\) is both the output of CNN and the input of GRU networks, and \(\mathbf{Y}_{G}\) is the output of GRU networks. Finally, the learned features are infinitely approximated to the final output of the network through the activation function. \[y_{i}=\text{PReLU}(x_{i}w_{i}+b_{i}), \tag{45}\]  
The PReLU [50] function has the characteristics of fast convergence and simple gradient calculation, and it is the activation function of the fully connected layer. \[\text{PReLU}(x_{x})= \begin{cases}x_{x}&x_{x}\geq 0\\ ax_{x}&\text{others}\end{cases} . \tag{46}\]  
In brief, the hybrid neural network consists of the convolutional layer, the max-pooling layer, the GRU layer, and the fully connected layer (Figure 3 and Table 1). For convenience, the capitalized initials of each layer are used to represent the corresponding network layer. The model first uses the convolutional layer C1 to learn features and then uses the Mish function to map the feature nonlinearly, where A1 is obtained. Next, the pooling layer P1 is applied to down-sample the features in A1. Then, the features in P1 are flattened into a vector and it is inputted into layer G2 for deep learning through GRU. Finally, the output is calculated through the full connection layer FC3. Figure 3: Flowchart of the proposed method. Understandably, the more training set data, the better the parameters of the fitting curve can be determined, which is more conducive to the final optimization and determination of the model. On the contrary, the more test set data, the more accurate the generalization error estimation of the model. Therefore, trade-offs need to be made when dividing the dataset. The dataset is divided based on the hold-out method. Generally, about 2/3\(\sim\)4/5 samples are used for training. The most common training set and test set proportions are 6:4, 7:3, or 8:2. As mentioned earlier, the data sets with different proportions of training sets (50%, 60%, 70%, 80%, and 90%) are trained and tested many times, and the average value is taken for comparison. It is found that the more training set data, the longer the training time, and the training time is 203 s-292 s. The tested root mean square errors (RMSEs) were 1.89%, 0.96%, 0.64%, 0.21%, and 0.26%, respectively. When the training set data accounts for 80% and 90%, the test RMSE is better, but the training time of 90% data is 292 s more than that of 80% data, which is 246 s. As mentioned earlier, we investigated the selection of the percentage of the training samples and finally determined that the training sample percentage is 80%. Additionally, we investigated the impact of batch size on the comprehensive ability of the model, and finally determined that the batch size is 64.