A study on a feedforward neural network to solve partial differential equations in hyperbolic-transport problems

TL;DR

This paper demonstrates that a straightforward deep neural network can effectively approximate solutions to complex hyperbolic PDEs, including shock waves and non-convex flux problems, showing promising accuracy compared to traditional methods.

Contribution

The study introduces a simple supervised deep learning approach for solving hyperbolic PDEs, successfully handling discontinuities and complex flux functions, which is a novel application in this context.

Findings

Achieved high-precision numerical solutions for shock and rarefaction waves.

Successfully modeled Buckley-Leverett two-phase flow with non-convex flux.

Deep learning solutions aligned well with classical and modern numerical methods.

Abstract

In this work we present an application of modern deep learning methodologies to the numerical solution of partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into account the equation and initial conditions of the model. We apply it to the Riemann problems over the inviscid nonlinear Burger's equation, whose solutions might develop discontinuity (shock wave) and rarefaction, as well as to the classical one-dimensional Buckley-Leverett two-phase problem. The Buckley-Leverett case is slightly more complex and interesting because it has a non-convex flux function with one inflection point. Our results suggest that a relatively simple deep learning model was capable of achieving promising results in such challenging tasks, providing numerical approximation of entropy solutions with very good precision and consistent to classical as well as to recently novel numerical methods in these particular scenarios.