## 2 Background Theory and Proposed Framework
### Proposed Framework for Predicting Train-Induced Bridge Responses
If the \(k_{\text{th}}\) axle load acts on the \(a_{\text{th}}\) node, the \(k_{\text{th}}\) axle load acts on each node at the \(n_{\text{th}}\) time step, as shown in Equation (14):  
\[\mathbf{\Pi}_{w}^{k}(i,\,1)= \begin{cases}\mathbf{A}_{\text{opt}}^{*}(k)&i\ =\ a\\ 0&i\ \
eq\ a\end{cases}  \tag{14}\]  
If the axle load operates between nodes \(a\) and \(a+1\), following the concept of equivalent load, the load can be partitioned into two components, as illustrated in Figure 2. The two components act independently on nodes \(a\) and \(a+1\), as expressed in Equation (15):  
\[\mathbf{\Pi}_{w}^{k}(i,\,1)= \begin{cases}m_{1}\times\mathbf{A}_{\text{opt}}^{*}(k)&i \ =\ a\\ m_{2}\times\mathbf{A}_{\text{opt}}^{*}(k)&i\ =\ a+1\\ 0&i\ \
eq\ a\text{ and }i\ \
eq\ a+1\end{cases}  \tag{15}\]  
where \(m_{1}\) and \(m_{2}\) are axle load distribution coefficients and can be calculated according to Equations (16) and (17):  
\[m_{2}\ =\ \mathbf{P}_{n}(k)/L_{e}-floor[\mathbf{P}_{n}(k)/L_{e}] \tag{16}\]  
\[m_{1}\ =\ 1-m_{2} \tag{17}\]  
In Equation (16), \(floor(\cdot)\) denotes the operation of rounding down a real number to the nearest integer. Subsequently, the load exerted on each structural node at the \(n_{\text{th}}\) time step can be determined, as depicted in Equation (18):  
\[\mathbf{\Pi}_{w}(\cdot,\ n)\ =\ \sum_{k=1}^{K^{*}}\mathbf{\Pi}_{w}^{k}(\cdot,\ 1) \tag{18}\]  
After the load on each node at each time step is calculated, the train load matrix \(\mathbf{\Pi}_{w}\) can be obtained. In the case of small- and medium-span railway bridges, the dynamic impact is relatively limited. Considering the scenario where the bridge experiences solely train load, the factors influencing the measured response of the bridge encompass bridge  
Figure 2: Equivalent decomposition of axial load. stiffness, bridge damping, bridge mass, track irregularity, bridge damage, train load, train speed, and sensor equipment performance, among others. For a specific railway bridge within a given timeframe, the properties or characteristics of the bridge structure and track irregularity are assumed to remain relatively stable. This stability establishes a consistent one-to-one nonlinear relationship between load input and response output. In this study, the load, train speed, and sampling time series of the train acting on each node are used as the input data of the neural network model \(\mathbf{X}_{i}\in\mathbb{R}^{N_{\mathbf{x}}\times N}\), and the bridge structure response is used as the output data \(\mathbf{Y}_{i}\in\mathbb{R}^{N_{\mathbf{y}}\times N}\). The dimension \(N_{\mathbf{x}}\) of the input data is \(N_{e}+3\), and the dimension \(N_{\mathbf{y}}\) of the output data is 2. The response output includes displacement \(\mathbf{R}_{\text{Disp}}\) and acceleration \(\mathbf{R}_{\text{Acc}}\), as shown in Equations (19) and (20):  
\[\mathbf{X}_{i}=\ [\mathbf{\Pi}_{\mathbf{w}};\ \mathbf{V};\ \mathbf{T}]\in\mathbb{R}^{(N_{e}+3) \times N} \tag{19}\]  
\[\mathbf{Y}_{i}=[\mathbf{R}_{\text{Disp}};\ \mathbf{R}_{\text{Acc}}]\in\mathbb{R}^{2 \times N} \tag{20}\]  
In order to predict the train-induced dynamic responses of a railway bridge, this paper establishes a LSTM deep neural network structure. This network structure includes normalization layer, data input layer, LSTM layer, dropout layer, full connection layer, regression output layer, denormalization layer, etc. Taking the two-layer stacked LSTM network as an example, the neural network model architecture is shown in Figure 3. In order to eliminate the dimensional influence between different input data \(\mathbf{X}_{i}\) and avoid the problem of gradient explosion, it is necessary to standardize the data to solve the comparability between data indicators and improve the speed of gradient descent and convergence. After the original data are processed by data standardization, each metric is in the same order of magnitude, that is, normalization.