## 8 Conclusions  
This study presents a novel PINN-based surrogate modeling framework for predicting the steady-state dynamic response of infinite E-B beams on foundations under moving loads across broad parameter ranges. Our methodology involves (1) constructing localized PINN models trained on displacement data from specific points in the damping-velocity parameter space and (2) employing transfer learning techniques to generalize these models across different parameter regions. The key innovation of our approach lies in its ability to maintain high prediction accuracy while substantially reducing data requirements compared with conventional PINN-based surrogate modeling methods. Numerical experiments demonstrate that when selecting 15 target domains, our method achieves comparable accuracy to traditional PINN surrogate models while requiring less than half the training data. For applications where minor accuracy reduction is acceptable, the number of target domains can be further reduced to 4.  
Empirical results confirm the proposed surrogate model's reliability and robustness under various noise conditions and transfer learning pathways. Notably, we find that simply increasing the number of target domains does not necessarily improve prediction accuracy. Instead, enhancing the spatial distribution density of target domains proves more effective for maintaining performance. These findings enable large-scale parameter prediction in data-limited scenarios while preserving computational efficiency. The proposed framework represents a significant advancement in data-efficient surrogate modeling, demonstrating particular effectiveness in high-accuracy prediction with minimal training data, robust performance across diverse parameter conditions, and adaptability to noisy measurement environments. This work establishes a foundation for efficient dynamic response prediction in scenarios where experimental data collection is challenging or costly.  
Existing PINN-based surrogate models often suffer from low training efficiency, primarily due to their dependence on extensive datasets. To address this limitation, we propose a novel method for predicting the steady-state response of infinite beams on foundations subjected to moving loads, particularly in scenarios where only sparse measurement data are available under constrained parameter conditions. A key practical challenge in our approach lies in balancing the trade-off between data requirements and predictive accuracy in surrogate modeling. Our numerical experiments demonstrate that partitioning the solution manifold into well-defined parameter regions enables the identification of optimal transfer learning pathways, thereby significantly improving model performance, especially in data-scarce regimes. Advancing this framework to optimize data efficiency and generalization capabilities will be the focus of our future research.  
Conceptualization, B.Z.; methodology, B.Z.; software, C.X. and L.O.; investigation, C.X. and L.O.; writing--original draft preparation, B.Z. and C.X.; writing--review and editing, B.Z. and L.O.; visualization, C.X.; supervision, B.Z.; project administration, B.Z.; funding acquisition, B.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China grant number 11672185.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
**Acknowledgments:** The authors would like to express their sincere gratitude to [PERSON] (China Construction Eighth Engineering Division Corp., Ltd., Shanghai, China) for his valuable guidance and assistance during the research and preparation of this paper.  
**Conflicts of Interest:** The authors declare no conflicts of interest.