An optimal control deep learning method to design artificial viscosities for Discontinuous Galerkin schemes

TL;DR

This paper introduces an optimal control deep learning approach to design artificial viscosities that effectively reduce non-physical oscillations in high-order Discontinuous Galerkin methods, improving solution stability and accuracy.

Contribution

It presents a novel neural network-based viscosity construction method formulated as an optimal control problem, leveraging gradient backpropagation for improved DG scheme performance.

Findings

Artificial viscosities outperform traditional methods

Neural network viscosities reduce oscillations effectively

Method achieves comparable or better results than existing approaches

Abstract

In this paper, we propose a method for constructing a neural network viscosity in order to reduce the non-physical oscillations generated by high-order Discontiuous Galerkin (DG) methods. To this end, the problem is reformulated as an optimal control problem for which the control is the viscosity function and the cost function involves comparison with a reference solution after several compositions of the scheme. The learning process is strongly based on gradient backpropagation tools. Numerical simulations show that the artificial viscosities constructed in this way are just as good or better than those used in the literatur