Physics Informed Neural Network for Solving the Shallow Water Equations on Grids of Arbitrary Geometry

Many physical phenomena exhibit wave behavior, including fluids, acoustics, and electromagnetics. When simulating wave behavior, one difficulty that is often encountered is wave reflections, which can lead to numerical oscillations especially on grids with complex geometries. In this research, we investigated if a neural network can accurately model wave behavior when extensive reflections are involved. If proven true, such a neural network would be very useful, as neural networks are amongst the fastest techniques for solving mathematical models. To test this idea, we trained a neural network to simulate the shallow water equations on a grid where the right boundary is fitted to a 5 point, 8 point, or 11 point spline. The simulation scenario called for a wave to be given an initial velocity in the right direction, causing it to impinge upon and then reflect off of the right boundary. To make the task even more difficult, we allowed the initial wave location to be a random point on the grid. Our training methodology involved both supervised learning with sample data points and unsupervised learning with the residuals of the shallow water equations acting as a loss function. Throughout the course of five experiments, the neural network learned to model wave reflection with a mean squared error of 9.7797E-06 for the simplest scenario and 1.6595E-05 for the most complex scenario. We have proven that a neural network is capable of learning wave behavior with reflections, with applications in computational acoustics to the modeling of detonation wave reflections.