## 6 Comparison of Result Accuracy with Existing PINN-Based Surrogate Model  
In contrast with existing PINN-based surrogate modeling techniques, our novel approach achieves a substantial reduction in data sampling requirements. Within a single source domain of the parameter space, merely 50 data points are sampled, spread over a 5 m span both ahead of and behind the moving load. Through the application of local PINN models, these 50 data points are augmented to 400. In Figures 12-14, we present the prediction accuracy demonstration of the surrogate model trained with the existing method based on more data [21, 22, 23, 32].  
Figure 11: Contour plot of (**a**) MAE, (**b**) RMSE, and (**c**) \(R^{2}\) values for our PINN surrogate model on the \(c-v\) parameter plane. The noise standard deviation is set to 3% of the true values’ amplitude.  
Figure 12: Contour plot of (**a**) MAE, (**b**) RMSE, and (**c**) \(R^{2}\) values for the existing PINN surrogate model on the \(c-v\) parameter plane. The noise standard deviation is set to 1% of the true values’ amplitude.  
Figure 13: Contour plot of (**a**) MAE, (**b**) RMSE, and (**c**) \(R^{2}\) values for the existing PINN surrogate model on the \(c-v\) parameter plane. The noise standard deviation is set to 2% of the true values’ amplitude.  
Compare Figures 12-14 with Figures 9-11. Although this augmentation process in our new approach may introduce a certain degree of inaccuracy when contrasted with measured values, it nonetheless maintains a satisfactory level of precision. Figure 15 illustrates a comparison of total data requirements and training time for a single target domain in the parameter space between our proposed method and the existing approach. The computer configuration utilized in our experiments is as follows: CPU AMD Ryzen 7 8845H with 8 cores, GPU NVIDIA GeForce RTX 4060 laptop, physical memory RAM 24 GB, and no GPU usage.  
In the transfer learning phase, when transferring the local PINN model from the source domain to the target domain, a portion of the network layers of the local PINN model are frozen. Consequently, the target domain marks a substantial reduction from the source domain. Overall, in comparison with existing methods, our approach diminishes measured data sampling by a minimum of 50%. Furthermore, when transferring the local PINN model from the source to the target domain, only a minor fine-tuning of weights is necessary to attain commendable accuracy. In our numerical experiments, the training efficiency improves by 15% in comparison with conventional methods for a single target domain in the parameter space. The extent of efficiency enhancement is contingent upon the number of target domains, which subsequently determines the final accuracy of the surrogate model. By moderately relaxing the local precision requirements, our method attains significant improvements in computational efficiency and a marked reduction in data demand. This strategic trade-off allows for a more efficient computational process while still maintaining a satisfactory level of accuracy.  
Figure 14: Contour plot of (**a**) MAE, (**b**) RMSE, and (**c**) \(R^{2}\) values for the existing PINN surrogate model on the \(c-v\) parameter plane. The noise standard deviation is set to 3% of the true values’ amplitude.  
Figure 15: Comparison of (**a**) the total training data requirements and (**b**) training time for a single target domain in the parameter space between the proposed method and existing methods.