A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network

In this work, we develop the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. Supervised learning is employed with the training data consisting of three-point stencils and the corresponding WENO3-JS weights as labels. We design two loss functions, one built on the mean squared error and the other from the mean squared logarithmic error. Each loss function consists of two components, where the first enforces the model to maintain the essentially non-oscillatory behavior while the second reduces the dissipation around discontinuities and improves the performance in smooth regions. We choose the shallow neural network (SNN) for computational efficiency with the Delta layer pre-processing the input. The resulting WENO3-SNN schemes outperform the classical WENO3-JS and WENO3-Z in one-dimensional examples, and show comparable sometimes superior simulations to WENO3-JS and WENO3-Z in two-dimensional examples.