PHYSICS-INFORMED NEURAL NETWORKS MODIFICATION FOR SOLVING 2D SHALLOW WATER EQUATIONS

Abstract

This study investigates the use of Physics-Informed Neural Networks (PINN) for solving the two-dimensional shallow water equations (SWE) on a flat bed and proposes a modified PINN with dynamic mesh-refinement strategy that adaptively increases the density of collocation points in critical or stiff regions such as propagating wave fronts and rapidly varying gradients. In the proposed framework, shallow-water physics is enforced as soft constraints in the loss, with initial and boundary conditions embedded to ensure a well-posed formulation without labelled targets. We evaluated fully connected architectures using the Adaptive Moment Estimation (Adam) and Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimisation algorithms.  Fully connected neural networks are trained using a combination of Adam and L-BFGS optimisers, and a FDM solution is employed as a reference for quantitative comparison. The standard PINN and its dynamically refined variant are evaluated against the FDM benchmark in terms of convergence behaviour and predictive accuracy. The results show that the refined PINN concentrates collocation points in stiff regions, reduces errors near wave fronts, and achieves accuracy closer to FDM with faster convergence.