A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.