  Recent research works for solving partial differential equations (PDEs) with
deep neural networks (DNNs) have demonstrated that spatiotemporal function
approximators defined by auto-differentiation are effective for approximating
nonlinear problems, e.g. the Burger's equation, heat conduction equations,
Allen-Cahn and other reaction-diffusion equations, and Navier-Stokes equation.
Meanwhile, researchers apply automatic differentiation in physics-informed
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 data-driven 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 auto-differentiation from traditional deep
learning framework, e.g. TensorFlow package, which improves the effectiveness
of capturing discontinuities in Riemann problems. In particular, the
2-Coarse-Grid neural network (2CGNN) and 2-Diffusion-Coefficient neural network
(2DCNN) are introduced in this work. We use 2 solutions of a conservation law
from a converging sequence, computed from a low-cost numerical scheme, and in a
domain of dependence of a space-time grid point as the input for a neural
network to predict its high-fidelity 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.
