Deep smoothness WENO scheme for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicators

TL;DR

This paper presents a deep learning-enhanced WENO scheme that improves shock-capturing accuracy in two-dimensional hyperbolic conservation laws by training neural networks to optimize smoothness indicators, outperforming traditional methods.

Contribution

The paper introduces a novel deep learning approach to improve WENO schemes by training neural networks for smoothness indicators without extra post-processing, enhancing shock resolution.

Findings

Enhanced accuracy near shocks in 2D Euler equations

Outperforms traditional WENO schemes in test cases

Reduces numerical diffusion and overshoot around shocks

Abstract

In this paper, we introduce an improved version of the fifth-order weighted essentially non-oscillatory (WENO) shock-capturing scheme by incorporating deep learning techniques. The established WENO algorithm is improved by training a compact neural network to adjust the smoothness indicators within the WENO scheme. This modification enhances the accuracy of the numerical results, particularly near abrupt shocks. Unlike previous deep learning-based methods, no additional post-processing steps are necessary for maintaining consistency. We demonstrate the superiority of our new approach using several examples from the literature for the two-dimensional Euler equations of gas dynamics. Through intensive study of these test problems, which involve various shocks and rarefaction waves, the new technique is shown to outperform traditional fifth-order WENO schemes, especially in cases where the numerical solutions exhibit excessive diffusion or overshoot around shocks.