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