## 5 The PINN-Based Surrogate Model
The model training employs an adaptive learning rate scheduling strategy, where the initial learning rate is set to \(1\times 10^{-3}\) and halved every 200 training epochs. A dual early-stopping mechanism is triggered when the learning rate decays below the \(1\times 10^{-6}\) threshold or when the validation loss shows no downward trend for 30 consecutive epochs. Additionally, the surrogate model's training employs the Adam optimizer. Training iterations are increased to 20,000. Notably, to prevent interference from sparsity constraints (e.g., L1/L2 regularization) with the physical governing equation loss terms, these regularization components have been intentionally excluded from the model design. In terms of prediction mechanism design, the surrogate model adopts a dynamic network activation strategy based on the domain features of the parameter space. More specifically, for a given prediction point (\(c^{*},v^{*}\)), the closest target-domain reference model within the parameter plane is first identified using the minimum Euclidean distance criterion, as follows:  
\[d_{i}=\sqrt{(c^{*}-c_{i})^{2}+(v^{*}-v_{i})^{2}},\,\,i\in[1,15]. \tag{34}\]  
The surrogate model activates the sub-network module corresponding to the nearest neighbor model. The target parameter coordinates (\(c^{*},v^{*}\)) and the beam coordinate \(y^{*}\) are then fed into this sub-network module simultaneously for deflection prediction. This mechanism significantly enhances the model's ability to represent local features by preserving the domain-specific characteristics inherent to the parameter space. To assess the model's generalization capability, this study creates a \(15\times 10\) grid of test points in the parameter space, defined by \(c\in[0.5,1,1.5,\dots,5]\times 10^{6}\) and \(v\in[10,15,20,\dots,80]\), totaling 150 test points. For each test point, 400 equidistant points are positioned within a 5 m span both ahead of and behind the moving load on the beam to predict deflection. We use the metrics MAE (mean absolute error), RMSE (root mean square error), and coefficient of determination \(R^{2}\) to evaluate the accuracy of model predictions. The formulations for MAE and RMSE are expressed as  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}|w_{\text{p}}^{i}(y_{j})-w_{\text{t}}^{i}(y_{j})| \tag{35}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(w_{\text{p}}^{i}(y_{j})-w_{\text{t}}^{i}(y _{j}))^{2}} \tag{36}\]  
where n is the number of sample data points, and \(w_{p}^{i}(y_{j})\) and \(w_{\text{t}}^{i}(y_{j})\) represent the predicted and actual displacement values, respectively. The evaluation values for each parameter point are mapped onto the \(c-v\) parameter plane, resulting in the contour plot shown in Figure 9. Figure 9 demonstrates the prediction accuracy of the surrogate models through three evaluation metrics (MAE, RMSE, and \(R^{2}\)). Numerical experiments reveal that, in most parameter regions, the MAE and RMSE values are approximately \(10^{-6}\), with \(R^{2}\) values approaching 1. Next, we incrementally increase the noise intensity by introducing controlled noise levels equivalent to 2% and 3% of the maximum amplitude of the analytical solutions. Figures 10 and 11 visually present the prediction results of the PINN surrogate model under higher noise standard deviation through MAE, RMSE, and \(R^{2}\). The figures show that as the noise standard deviation increase, MAE and RMSE values for extreme outliers rise; however, in most parameter regions of the model, MAE and