Eno classification and regression neural networks for numerical approximation of discontinuous flow problems

Learning high order non-oscillatory polynomial approximation procedures which form the backbone of high order numerical solution of partial differential equations is challenging. The major issue is to pose these procedures as a learning problem and generate suitable synthetic data set which suffice learning it with small neural networks. In this work, we pose an arc-length based essentially non-oscillatory (ENOL) reconstruction algorithm as machine learning problem. A novel way to construct the synthetic data using ENOL algorithm along with basic smooth and piece-wise continuous functions is given. Small vanilla regression and classification neural networks are trained to learn third order (ENOL) polynomial approximation procedure. The metric of trained ENO classification and regression networks is presented and commented. These trained models are implemented in numerical solver to compute the solution of test problems of hyperbolic conservation laws. The presented numerical results show that ENO classification network gives results comparable to the exact ENOL reconstruction whereas ENO regression network performs poorly both in terms of convergence and resolving the discontinuities.