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Recent research works for solving partial differential equations (PDEs) withdeep neural networks (DNNs) have demonstrated that spatiotemporal functionapproximators defined by auto-differentiation are effective for approximatingnonlinear 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-informedneural network (PINN) to solve nonlinear hyperbolic systems based onconservation laws with highly discontinuous transition, such as Riemannproblem, by inverse problem formulation in data-driven approach. However, itremains a challenge for forward methods using DNNs without knowing part of thesolution to resolve discontinuities in nonlinear conservation laws. In thisstudy, we incorporate 1st order numerical schemes into DNNs to set up the lossfunctional approximator instead of auto-differentiation from traditional deeplearning framework, e.g. TensorFlow package, which improves the effectivenessof capturing discontinuities in Riemann problems. In particular, the2-Coarse-Grid neural network (2CGNN) and 2-Diffusion-Coefficient neural network(2DCNN) are introduced in this work. We use 2 solutions of a conservation lawfrom a converging sequence, computed from a low-cost numerical scheme, and in adomain of dependence of a space-time grid point as the input for a neuralnetwork to predict its high-fidelity solution at the grid point. Despitesmeared input solutions, they output sharp approximations to solutionscontaining shocks and contacts and are efficient to use once trained.

Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems