Physics‐Informed Neural Networks for Solving the Two‐Dimensional Shallow Water Equations With Terrain Topography and Rainfall Source Terms

Solving the two‐dimensional Shallow Water Equations (SWE) is a fundamental problem in flood simulation technology. In recent years, physics‐informed neural networks (PINNs) have emerged as a novel methodology for addressing this problem. Given their advantages in parallel computing, potential for data assimilation and parameter calibration, and the rapid advancement of artificial intelligence, it is crucial to investigate both the capabilities and limitations of PINNs. While current research has demonstrated the significant potential of PINNs, many aspects of this new approach remain to be explored. In this paper, we employ PINNs enhanced by dimensional transformation and Neuron‐wise Local Adaptive Activation Function techniques to validate its effectiveness in solving two‐dimensional free surface flow with rainfall on terrain topography. The SWE primarily exist in two forms: the primitive form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid primitive‐conservative form offers superior performance. Additionally, in our cases, we find that incorporating the energy conservation law, specifically the entropy condition, does not improve accuracy and sometimes destabilizes training due to an over‐weighting of the mass conservation law. Furthermore, we have developed an open‐source module on the PINNacle platform for solving SWE using PINNs, which includes over 10 case studies and various equation forms, to promote research and application in this field.