An improved physics-informed neural network for solving shallow water equations

The shallow water equations are the nonlinear hyperbolic partial differential equations often discussed in Computational Fluid Dynamics (CFD). The nonlinear nature of the equations makes it difficult to obtain analytical solutions. Besides, the solutions may develop into discontinuity even if the initial conditions are smooth. Therefore, to solve shallow water equations precisely has always been a hot topic. The classic Physics-informed Neural Networks (PINNs) have some shortcomings when solving shallow water equations and usually resulting in unsatisfactory approximation near discontinuities, e.g. significant numerical dissipation or pseudo oscillation. In order to capture the discontinuities more accurately, we propose an improved PINN to solve shallow water equations. The improved PINN combines the domain extension algorithm and the weighting strategy. The domain extension algorithm allows the network to make predictions over a wider area and to better handle boundary conditions and initial conditions, thereby improving its ability to capture discontinuities. The weighting strategy assigns different weights to various physical constraints, which allows the network to be more flexible on physical laws. We present three numerical experiments, namely, dam break problem, dam break problem with near-zero water depths and flow on the hump. By comparing the results with those obtained by the fifth-order WENO-Z scheme, the validity and high accuracy of the improved PINN are proved.