Solution to the wave equation using physics-informed machine learning

Abstract

We developed and optimised an artificial neural network (ANN) solution for the linear homogeneous hyperbolic wave partial differential equation (PDE) and analysed its errors against an approximated finite difference method (FDM) solution. The wave PDE formulated has constant coefficients and is well-posed with physical initial and boundary conditions. The ANN formulated has a fully-connected feed-forward architecture. The PDE residual and conditions are embedded into the ANN solution on rectangular domains through physics-informed machine learning, gradient-based optimisation, regularisation, and hyperparameter tuning. We found that the ANN solution achieved lower errors, demonstrating the potential of physics-informed machine learning models as better and flexible approximators to PDE solutions for modelling physical phenomena.