标题： 基于物理信息的神经网络求解具有地形高程和降雨源项的二维浅水方程 显示英文标题

标题： Physics-Informed Neural Networks for Solving the Two-Dimensional Shallow Water Equations with Terrain Topography and Rainfall Source Terms

摘要： 求解二维浅水方程是洪水模拟技术中的基本问题。 近年来，物理信息神经网络（PINNs）作为一种新方法被提出，用于解决这个问题。 鉴于其在并行计算方面的优势、数据同化和参数校准的潜力以及人工智能的快速发展，研究PINNs的能力和局限性至关重要。 尽管当前研究已经展示了PINNs的巨大潜力，但这一新方法的许多方面仍有待探索。 在本研究中，我们采用通过维度变换和N-LAAF技术增强的PINNs，以验证其在地形地貌上降雨引起的二维自由表面流中的有效性。 浅水方程主要存在两种形式：变量形式和守恒形式。 通过理论分析和实验验证，我们证明了一种混合变量-守恒形式具有更好的性能。 此外，我们发现引入能量守恒定律，特别是熵条件，并不会带来显著的改进，甚至可能导致训练失败。 此外，我们已经在PINNacle平台上开发了一个开源模块，用于使用PINNs求解浅水方程，该模块包含十余个案例研究和各种方程形式，以促进该领域的研究和应用。 显示英文摘要

摘要： Solving the two-dimensional shallow water equations 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, the 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 study, we employ PINNs enhanced by dimensional transformation and N-LAAF techniques to validate their effectiveness in solving two-dimensional free surface flow with rainfall on terrain topography. The shallow water equations primarily exist in two forms: the variables form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid variable-conservation form offers superior performance. Additionally, we find that incorporating the energy conservation law, specifically the entropy condition, does not yield substantial improvements and may even lead to training failure. Furthermore, we have developed an open-source module on the PINNacle platform for solving shallow water equations using PINNs, which includes over ten case studies and various equation forms, to promote research and application in this field.