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

TL;DR

This paper presents a neural network-based third-order finite difference WENO scheme that improves accuracy and efficiency for hyperbolic conservation laws, outperforming traditional methods in 1D and 2D tests.

Contribution

The paper introduces a shallow neural network to enhance WENO schemes, eliminating post-processing and improving performance around discontinuities.

Findings

Outperforms traditional WENO3-JS and WENO3-Z schemes in 1D tests.

Shows improved behavior in 2D simulations.

Uses a shallow neural network for computational efficiency.

Abstract

In this paper, we introduce the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. We employ the supervised learning and design two loss functions, one with the mean squared error and the other with the mean squared logarithmic error, where the WENO3-JS weights are computed as the labels. Each loss function consists of two components where the first component compares the difference between the weights from the neural network and WENO3-JS weights, while the second component matches the output weights of the neural network and the linear weights. The former of the loss function enforces the neural network to follow the WENO properties, implying that there is no need for the post-processing layer. Additionally the latter leads to better performance around discontinuities. As a neural network structure, we choose the shallow neural network (SNN) for computational efficiency with the Delta layer consisting of the normalized undivided differences. These constructed WENO3-SNN schemes show the outperformed results in one-dimensional examples and improved behavior in two-dimensional examples, compared with the simulations from WENO3-JS and WENO3-Z.