Enhanced fifth order WENO Shock-Capturing Schemes with Deep Learning

TL;DR

This paper introduces a deep learning-enhanced fifth order WENO scheme that improves shock-capturing accuracy without additional post-processing, with proven convergence and superior performance on classical test problems.

Contribution

The paper presents a novel integration of deep learning into WENO schemes, specifically training a neural network to modify smoothness indicators, enhancing shock-capturing capabilities.

Findings

Outperforms classical WENO in shock resolution

No additional post-processing needed for consistency

Convergence of the scheme is theoretically proven

Abstract

In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method, which simplifies the method and increases the effect of the neural network. Moreover, the convergence of the resulting scheme can be theoretically proven. We demonstrate our findings with the inviscid Burgers' equation, the Buckley-Leverett equation and the 1-D Euler equations of gas dynamics. Hereby we investigate the classical Sod problem and the Lax problem and show that our novel method outperforms the classical fifth order WENO schemes in simulations where the numerical solution is too diffusive or tends to overshoot at shocks.