Controlling spurious numerical oscillations using neural networks

Abstract: High-order methods for approximating solutions to hyperbolic systems of  conservation laws, need to be carefully treated near discontinuities to avoid Gibbs oscillations.  In the context of discontinuous Galerkin (DG) schemes, several methods have been developed to control spurious oscillations,  such as limiting the local  approximating polynomials or adding shock capturing terms.  However, most existing  methods require the prescription of problem-dependent parameters  which are usually determined empirically.  A non-optimal choice of these parameters can either  lead to the re-appearance of Gibbs oscillations, or the loss of accuracy in smooth regions.                                                                         

In this talk, we propose a new data-driven approach to overcome this bottleneck. In particular, we train artificial neural networks (ANNs) using supervised learning,  which are then used as a black-box to detect numerical discontinuities and prescribe artificial viscosity required in the shock capturing terms.  The advantage of the proposed strategy is that it is parameter-free, computationally efficient,  and can easily be integrated into existing code frameworks. Several numerical results are presented to demonstrate the robustness of the networks in the framework of Runge-Kutta DG schemes.  This work was done jointly with Jan S. Hesthaven and Niccolò Discacciati.