Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems
Abstract
Recent research works for solving partial differential equations (PDEs) with deep neural networks (DNNs) have demonstrated that spatiotemporal function approximators defined by autodifferentiation are effective for approximating nonlinear problems, e.g. the Burger's equation, heat conduction equations, AllenCahn and other reactiondiffusion equations, and NavierStokes equation. Meanwhile, researchers apply automatic differentiation in physicsinformed neural network (PINN) to solve nonlinear hyperbolic systems based on conservation laws with highly discontinuous transition, such as Riemann problem, by inverse problem formulation in datadriven approach. However, it remains a challenge for forward methods using DNNs without knowing part of the solution to resolve discontinuities in nonlinear conservation laws. In this study, we incorporate 1st order numerical schemes into DNNs to set up the loss functional approximator instead of autodifferentiation from traditional deep learning framework, e.g. TensorFlow package, which improves the effectiveness of capturing discontinuities in Riemann problems. In particular, the 2CoarseGrid neural network (2CGNN) and 2DiffusionCoefficient neural network (2DCNN) are introduced in this work. We use 2 solutions of a conservation law from a converging sequence, computed from a lowcost numerical scheme, and in a domain of dependence of a spacetime grid point as the input for a neural network to predict its highfidelity solution at the grid point. Despite smeared input solutions, they output sharp approximations to solutions containing shocks and contacts and are efficient to use once trained.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.09316
 arXiv:
 arXiv:2109.09316
 Bibcode:
 2021arXiv210909316H
 Keywords:

 Mathematics  Numerical Analysis;
 Physics  Fluid Dynamics