## Case Study  
### Engineering Background  
A simply supported concrete box girder bridge with a span of 32 m was taken as the research object. The elastic modulus of the selected reinforced concrete material, denoted as \(E\), is \(3.45\times 10^{10}\) N/m\({}^{2}\). The section area, denoted as \(A\), is \(8.89\) m\({}^{2}\). The moment of inertia of section, denoted as \(I_{y}\), is \(10.95\) m\({}^{4}\). Based on the modal analysis, the first and second natural frequencies of the target bridge are obtained, i.e., \(5.12\) Hz and \(17.04\) Hz, respectively. The bridge was divided into \(N_{e}\) structural elements; thus, \(N_{e}\) structural nodes can be obtained in the established finite element model.  
The German Intercity-Express (ICE) multiple unit train, comprising two power cars and several trailers, is taken as the research object. The power cars are positioned at the front and rear of the train, while the trailers are situated in the middle. The axle spacing dimensions of both the trailers and power cars in each train formation are standardized, ensuring consistent fixed axle spacing \(l_{1}\), vehicle pitch \(l_{2}\), and car body length \(l_{3}\) within the same train. The individual axle loads of the power cars and trailers are \(16\) tons and \(14.6\) tons, respectively. The vehicle pitch \(l_{2}\) is \(17.375\) m, and the length of the carriage \(l_{3}\) is \(24.775\) m. The German low-disruption track vertical irregularity spectrum is employed as the input excitation. The spatial frequency \(\Omega\) ranges from \(0.05\) to \(3\) rad/m. The random spatial frequency is \(500\). The layout of the train, track, and bridge system is illustrated in detail in Figure 5.  
Series finite element simulations were carried out to obtain the dataset of train loads and structural responses for training the LSTM network. The train loads include the time history of axle loads, axle spacings, and axle positions. The structural responses include the displacement and acceleration of the bridge. The finite element model of the bridge comprises \(300\) units. The LSTM network takes train loads, train velocity, and sampling time series as inputs and vertical displacement and acceleration responses at the mid-span section of the bridge as outputs.  
This section aims to investigate the influencing factors, including the neural network structure, training sample size, number of bridge units, input noise, train speed, and train composition, on the prediction accuracy of train-induced responses utilizing the LSTM network. Initially, this study investigates the impact of the number of LSTM layers and units in hidden layers on the prediction accuracy. The dataset used for training neural network model is defined as dataset C, comprising \(100\) samples covering various train speeds, train compositions, train spacings, and sampling frequencies, as tabulated in Table 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Dataset Name**} & **Train** & **Number of** & **Fixed Wheel** & **Sampling** \\  & **Speed/(km/h)** & **Carriages** & **Base \(l_{1}\)/m** & **Frequency/Hz** \\ \hline A: sample size is 25 & 20, 40, 60 & 4, 8, 12 & 1.8, 2.5 & 100, 200 \\ \hline B: sample size is 50 & 20, 40, 60, 80 & 4, 8, 12, 16 & 1.8, 2.5 & 100, 200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of the training datasets.  
Figure 5: The layout of the train, track, and bridge system.  
The accuracy of predicted bridge responses using the same neural network structures with different data sizes are calculated. The training dataset sizes for the neural network range from 25 to 200, denoted as A to E, respectively. In addition, the influence of other factors, i.e., train speed, the number of carriages, and axel space, are also investigated. The detailed information for these datasets is provided in Table 1.