Discovering Artificial Viscosity Models for Discontinuous Galerkin Approximation of Conservation Laws using Physics-Informed Machine Learning

TL;DR

This paper introduces a physics-informed machine learning approach to automatically discover artificial viscosity models for discontinuous Galerkin methods solving conservation laws, improving accuracy near discontinuities without supervised data.

Contribution

It presents a reinforcement learning-inspired neural network algorithm that trains viscosity models in a dataset-free manner and demonstrates superior performance and generalization in numerical tests.

Findings

The learned viscosity model outperforms classical models.

The approach generalizes across different problems and parameters.

The method integrates seamlessly with existing DG solvers.

Abstract

Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models in a non-supervised paradigm. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.