Learning shallow water equations with physics-informed Deep Operator Network (DeepONet)

Physics-informed neural networks are an optimization-based approach for solving differential equations and have the potential to significantly speed up the modelling of complex phenomena, which conventionally is achieved via expensive numerical solvers. We present a Physics-Informed Deep Operator Network (DeepONet) framework for solving two-dimensional shallow water equations with variable bed topography under given boundary and initial conditions. While traditional physics-informed neural networks can solve differential equations on meshless grids using prescribed conditions, they require retraining for each new set of initial and boundary conditions. Our approach uses a DeepONet to learn the underlying solution operator rather than individual solutions, which provides an enhanced generalizability, making the DeepONet a feasible candidate for real world applications. The framework combines the advantages of neural networks with physical laws, effectively handling the complexities of varying bed topography and wet-dry transitions. We demonstrate that our DeepONet approach achieves comparable accuracy to classical numerical methods while significantly reducing inference time. In our modelling experiments we investigate the sensitivity of hyperparameter values and network architecture as well as the potential of introducing an additional data loss, emulating the availability of additional observational data on water levels or inundation extent.  This acceleration in computation speed makes the method particularly valuable for time-critical applications such as flood forecasting. The results establish physics-informed DeepONets as a promising alternative to traditional numerical solvers for shallow water systems, offering a balance between computational efficiency and solution accuracy.