Convolution Neural Network Shock Detector for Numerical Solution of Conservation Laws

Abstract

We propose a universal discontinuity detector using convolution neural network (CNN) and apply it in conjunction of solving nonlinear conservation laws in both 1D and 2D. The CNN detector is trained offline with synthetic data. The training data are generated using randomly constructed piecewise functions, which are then processed using randomized linear advection solver to count for the cases of numerical errors in practice. The detector is then paired with high-order numerical solvers. In particular, we combined high-order WENO in troubled cells with high-order central difference in smooth region. Extensive numerical examples are presented. We observe that the proposed method produces notably sharper and cleaner signals near the discontinuities, when compared to other well known troubled cell detector methods.