Accelerating high order discontinuous Galerkin solvers using neural networks: 1D Burgers’ equation

Abstract

High order discontinuous Galerkin methods allow accurate solutions through the use of high order polynomials inside each mesh element. Increasing the polynomial order leads to high accuracy, but increases the cost. On the one hand, high order polynomials require more restrictive time steps when using explicit temporal schemes, and on the other hand the quadrature rules lead to more costly evaluations per iteration. We propose to accelerate high order discontinuous Galerkin methods using Neural Networks. To this aim, we train a Neural Network using a high order discretisation, to extract a corrective forcing that can be applied to a low order solution with the aim of recovering high order accuracy. With this corrective forcing term, we can run a low order solution (low cost) and correct the solution to obtain high order accuracy. We provide error bounds to quantify the various errors included in the methodology (e.g. related to the discretisation or the Neural Network) . The methodology and bounds are examined for a variety of meshes, polynomial orders and viscosity values for the 1D viscous Burgers’ equation. The result show good accuracy and accelerations specially when considering high polynomial orders.