## 3 Construction and Training of Local PINN Models with Limited Data
### Implementation Details  
This subsection develops local PINN models to learn from sensor measurement data. Three specific points are selected in the damping-velocity \((c-v)\) parameter plane (Figure 2a): \((c,v)=\{(2.0\times 10^{6},10),(2.25\times 10^{6},10),(2.5\times 10^{6},10)\}\). The remaining parameters are derived from Equation (25) in Section 2, while the magnitude of the moving load is adjusted to \(P=10^{3}(kN)\). A separate PINN model is constructed for each parameter point, yielding three models in total. Because damping ensures that the vertical displacement approaches zero far from the moving load, only finite-region displacement data near the load is required for testing. Here, 20 equidistant points and their displacements are sampled within a 5 m span upstream and downstream of the load, forming the model's input dataset. Figure 3: Schematic diagram of local PINN framework. The output predicts displacements at any coordinate in the domain. To emulate real-world measurement noise, Gaussian white noise (with a standard deviation of 1% of the analytical solution's maximum amplitude) is superimposed on the analytical deflection values from Equations (27) and (28) to generate training data. After preliminary exploration, we systematically optimize the neural network architecture by determining the optimal hyperparameters for hidden layers and neurons per layer. Six distinct network configurations are evaluated through comprehensive trial computations, with detailed specifications provided in Table 1. The convergence characteristics of each configuration throughout the training process are illustrated in Figure 4.  
Figure 4: The training convergence curves for all six network topologies (\(S_{1}-S_{6}\)) defined in Table 1.  
Figure 4 demonstrates a consistent pattern: model accuracy improves with additional hidden layers, but at the cost of increased overfitting risk. Networks with more layers show greater oscillation during convergence, a characteristic sign of overfitting. Through systematic evaluation of convergence rate, oscillation magnitude, and prediction accuracy, we identify an optimal architecture of three hidden layers with 20 neurons each. This configuration achieves robust accuracy while maintaining stability and generalizability. For consistency, we employ this identical architecture (\(3\times 20\) neurons) across all subsequent transfer learning and surrogate models. All three local PINNs train using the Adam optimizer with an initial learning rate of 0.01. The 2000-iteration training process implements learning rate decay every 100 iterations, terminating when either (1) the learning rate decreases to \(10^{-6}\) or (2) the validation loss remains unchanged for 100 consecutive iterations. This strategy effectively prevents overfitting. Figure 5 shows the models' predicted deflection values at all three parameter points exhibiting excellent agreement with analytical solutions (Equations (27) and (28)), confirming high predictive accuracy. To more rigorously assess the model's predictive accuracy, we employ the coefficient of determination (\(R^{2}\)) as an evaluation metric. The calculation formula is as follows:  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(\tilde{\sigma_{i}}-\sigma_{i})^{2}}{\sum_{i=1}^{n }(\tilde{\sigma}-\sigma_{i})^{2}} \tag{31}\]  
where \(n\): number of sample data points; \(\sigma_{i}\): observed (true) values; \(\tilde{\sigma_{i}}\): model-predicted values; \(\tilde{\sigma}\): mean of observed values; \(\sum_{i=1}^{n}(\tilde{\sigma_{i}}-\sigma_{i})^{2}\): residual sum of squares (RSS), quantifying prediction errors; \(\sum_{i=1}^{n}(\tilde{\sigma}-\sigma_{i})^{2}\): total sum of squares (TSS), measuring data variance. The coefficient of determination (\(R^{2}\)) evaluates model performance, \(R^{2}\to 1\) indicates superior predictive accuracy, and \(R^{2}\to 0\) reflects poorer performance. Table 2 presents the \(R^{2}\) values for the PINN models when learning displacement data across three parameter points.