Learning 2D Shallow Water Equations With Physics‐Informed Neural Operator Networks

Abstract

This study investigates the application of Physics‐Informed Neural Operators (PINOs) for solving the two‐dimensional shallow water equations (2D SWE) in the context of flood modeling. Unlike Physics‐Informed Neural Networks (PINNs), which require retraining for each new initial or boundary condition (BC), PINOs learn the solution operator itself, enabling fast inference across a range of conditions without retraining. PINNs have moreover been shown to struggle with complex bed topography due to the difficulty of satisfying the well‐balanced property (Tian et al., 2025, https://doi.org/10.1029/2025wr040052 ; Dazzi, 2024, https://doi.org/10.1029/2023wr036589 ), a limitation that PINOs can partially mitigate by incorporating data alongside the physics loss. We evaluate the method on experiments of increasing physical complexity: a radial dam break, constant BCs with and without friction, time‐dependent BCs, and a real‐world test case. The results show that PINOs capture key flood dynamics, particularly water depth, while reducing inference time by up to two orders of magnitude compared to numerical solvers. Relative test errors for water depth ranged from 0.3% for the radial dam break to 10.9% for cases with bottom topography, 7.3% with friction, and 9.0% under time‐dependent BCs. For the real‐world case, an additional data loss term yielded a water depth error of 25.8%. Larger errors were consistently observed for velocity components. The findings establish PINOs as a promising complement to traditional numerical solvers, offering a balance between computational efficiency and solution accuracy. Future work will focus on improving accuracy and extending the framework toward real‐world flood forecasting applications.