An Optimal Control Deep Learning Method to Design Artificial Viscosities for Discontinuous Galerkin Schemes

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 Discontinuous Galerkin methods on uniform Cartesian grids. 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, with a convolutional architecture, constructed in this way are just as good or better than those used in the literature.