## 7 Discussion  
In this section, we conduct a systematic analysis of how migration path configurations impact the predictive accuracy of proposed PINN-based surrogate models. By examining the interactive effects of varying migration paths and noise levels on model performance, this evaluation provides critical insights into the reliability and robustness of the proposed surrogate models across diverse operational scenarios. Our findings deepen the understanding of how these factors collectively influence predictive capabilities and establish a foundation for optimizing transfer learning strategies in practical applications. We consider the following four distinct migration paths depicted in Figure 16:  
* Figure 16a: Derived from the configuration in Figure 7, this path reduces the number of target domains from 15 to 11, with all target domains positioned above the demarcation curve in the parameter plane. * Figure 16b: Building on Figure 7, the number of target domains is reduced from 15 to 9, and the target domains are distributed across both sides of the demarcation curve. * Figure 16c: Extending from Figure 16b, the number of target domains is further reduced from 9 to 4, with both the source and target domains positioned below the demarcation curve. * Figure 16d: Derived from Figure 7, this path reduces the number of source domains from 3 to 2 while decreasing the number of target domains from 15 to 10. These configurations collectively allow for a thorough assessment of how transfer learning strategies perform across different levels of data sparsity and domain positioning. This evaluation provides critical insights into enhancing model reliability and robustness in practical applications. Figure 16: Schematic diagram of four distinct migration paths on the \(c-v\) parameter plane: (**a**) 11 target domains, (**b**) 9 target domains, (**c**) 4 target domains, and (**d**) 10 target domains. Arrows represent the direction of migration paths. Figures 17-19 depict how the four distinct migration paths illustrated in Figure 16 affect the surrogate model's prediction accuracy under varying noise intensities, with noise standard deviations set at 1%, 2%, and 3% of the true value's amplitude. Figures 17-19 show that the transfer learning strategy on the \(c-v\) parameter plane significantly impacts the surrogate model's prediction accuracy. Under the same noise conditions, reducing the number of target domains leads to a corresponding drop in prediction accuracy. Moreover, when the number of target domains is progressively reduced while their distribution density in the occupied space remains largely unchanged, the surrogate model's prediction accuracy consistently declines across the entire parameter plane. Conversely, when the distribution density of target domains in the occupied space is significantly reduced, the decline in prediction accuracy becomes notably non-uniform across the parameter plane, with some regions experiencing greater accuracy reductions than others (Figures 17-1, 18-1, and 19-1). This spatially heterogeneous accuracy reduction underscores the importance of both the number and spatial organization of target domains in determining the surrogate model's performance. To enhance the surrogate model's reliability and robustness, particularly in complex modeling scenarios where data sparsity may be a concern, it is crucial to maintain adequate distribution density of target domains and strategically optimize their spatial arrangement. Figure 17: _Cont._  
Figure 17: The impact of different migration paths on the prediction accuracy of the proposed surrogate model illustrated through three evaluation metrics, MAE, RMSE, and \(R^{2}\). The noise standard deviation is 1% of the true value’s amplitude. Figure 18: The impact of different migration paths on the prediction accuracy of the proposed surrogate model illustrated through three evaluation metrics, MAE, RMSE, and \(R^{2}\). The noise standard deviation is 2% of the true value’s amplitude.