Learning 2D Shallow Water Equations with Physics-Informed Neural Operator Networks

Our study explores the use of Physics-Informed Neural Operators (PINOs) for solving the two-dimensional shallow water equations (2D SWE) in the context of flood modeling. In contrast to Physics-Informed Neural Networks (PINNs), which require retraining for each new set of initial or boundary conditions, PINOs learn the underlying solution operator, enabling rapid inference across a wide range of conditions without retraining.

We assess the performance of PINOs through a sequence of numerical experiments with increasing physical complexity, including a radial dam-break scenario, constant boundary conditions with and without friction, and time-dependent boundary conditions. Existing PINO frameworks were adapted and extended to accommodate these experimental settings.

The results demonstrate that PINOs can accurately capture key flood-relevant dynamics, particularly water depth, while achieving substantial computational speed-ups of up to four orders of magnitude compared to conventional numerical solvers. Relative test errors for water depth were as low as 0.3% for the radial dam-break case, increasing to 10.9% in the presence of bottom topography, 7.3% with friction, and 9.0% under time-dependent boundary conditions. Larger errors were observed for the velocity components.

The combination of competitive accuracy and significant computational acceleration highlights the potential of PINOs for time-critical applications such as flood forecasting. Overall, this work positions PINOs as a promising alternative to traditional numerical solvers for the 2D SWE, offering an effective balance between computational efficiency and solution fidelity. Future research will focus on improving predictive accuracy, expanding the diversity of training functions, and enhancing applicability to real-world flood scenarios.