## 2 Background Theory and Proposed Framework
### Proposed Framework for Predicting Train-Induced Bridge Responses  
This section introduces the proposed framework for predicting train-induced bridge dynamic responses based on the designed LSTM network according to the vehicle-bridge coupling theory. According to the theory of structural dynamics, the vehicle-bridge coupling system is a multi-degree of freedom system. By solving the dynamic system equation [20], the response information of displacement, velocity, and acceleration of each degree of freedom can be obtained. Two types of information are required to solve the dynamic equation. The first one is the physical properties of the vehicle-bridge system, including the mass matrix \(M\), the damping matrix \(C\), and the stiffness matrix \(K\). The second is the load information of the vehicle-bridge system, including the load actioning on the bridge and the vehicle at any time. In short, the vehicle-bridge coupling equation constructs the nonlinear relationship between vehicle load and bridge responses according to the physical law of structural dynamics. Therefore, this study considers the load of each node of the bridge structure at each time as the input information to the neural network, so as to establish an LSTM network that predicts the train-induced bridge responses. During the process of the train progressing from entering the bridge to leaving the bridge, the sampling numbers and frequency are \(N\) and \(f\) respectively, the speed is \(\mathbf{V}=[V_{1}~{}V_{2}~{}V_{3}\ldots V_{N}]\in\mathbb{R}^{1\times N}\), and the bridge span is \(L\). Usually, the axle loads can be identified before the train enters the target bridge using some load identification methods and equipment [21]. Additionally, the initial axle position of the train (the axle position corresponding to the initial value of the response time series) \(\mathbf{P}^{*}_{\text{opt}}\in\mathbb{R}^{1\times K^{*}}\) and the axle weight estimation value \(\mathbf{A}^{*}_{\text{opt}}\in\mathbb{R}^{K^{*}\times 1}\) can be obtained from the structural-health-monitoring system. The bridge structure is divided into \(N_{e}\) unit structures on average, that is, there are \(N_{e}+1\) structural nodes, and the length of each unit structure is shown in Equation (12):  
\[L_{e}~{}=~{}L/N_{e} \tag{12}\]  
Under the assumption of a singular train traversing the bridge, with pertinent information regarding the train load, bridge structure, and sampling protocol available, it becomes feasible to derive the load value at any longitudinal position on the bridge structure at any given time during the data acquisition process. The neural network's input data encompass the train load, train speed, and the sampling frequency of the sensor, while the output data comprise dynamic response information pertaining to the train. The input data \(\mathbf{X}_{i}\in\mathbb{R}^{N_{e}\times M_{i}}\) and the output data \(\mathbf{Y}_{i}\in\mathbb{R}^{N_{y}\times M_{i}}\) of the \(i_{\text{th}}\) group data sample can be constructed as follows. Assume the collected data of a train riding through the bridge as the \(i_{\text{th}}\) input and output pairs. The time step of the \(i_{\text{th}}\) group of data samples is \(M_{i}=N\), time series \(\mathbf{T}=[0,\frac{1}{7},\frac{2}{7},\ldots,\frac{n-1}{f},\frac{n}{7},\ldots,\frac {N-1}{f}]\in\mathbb{R}^{N\times 1}\). At the \(n_{\text{th}}\) time step \(\mathbf{T}(n)\), the train position on the bridge, denoted as \(\mathbf{P}_{n}\in\mathbb{R}^{1\times K^{*}}\), can be expressed as Equation (13):  
\[\mathbf{P}_{n}\ =\ \mathbf{P}_{\text{opt}}^{*}+\sum_{i=1}^{n-1}\frac{1}{f}(V_{n}+V_{n+1 })/2 \tag{13}\]  
The train load matrix is defined as \(\mathbf{\Pi}_{w}\in\mathbb{R}^{(N_{e}+1)\times N}\), where the element of row \(i\) and column \(j\) represents the axle load value at the time step \(j\) of the \(i_{\text{th}}\) structural node. The axle load of the \(k_{\text{th}}\) (\(k=1,\,2,\,3,\,\ldots,\,K^{*}\)) axle is \(\mathbf{A}_{\text{opt}}^{*}(k)\), and the action position of \(\mathbf{T}(n)\) is \(\mathbf{P}_{n}(k)\).