Conservative approximation-based feedforward neural network for WENO schemes

TL;DR

This paper introduces a neural network-based approach to improve WENO schemes for solving hyperbolic conservation laws, replacing classical weighting with a learned conservative approximation that enhances accuracy and robustness.

Contribution

The authors develop a neural network that replaces traditional WENO weights, trained with a novel loss function to ensure conservative approximation and symmetry, leading to improved scheme performance.

Findings

WENO3-CADNNs outperform WENO3-Z in benchmarks.

The neural network achieves accuracy comparable to WENO5-JS.

The approach demonstrates robust generalization across scenarios.

Abstract

In this work, we present the feedforward neural network based on the conservative approximation to the derivative from point values, for the weighted essentially non-oscillatory (WENO) schemes in solving hyperbolic conservation laws. The feedforward neural network, whose inputs are point values from the three-point stencil and outputs are two nonlinear weights, takes the place of the classical WENO weighting procedure. For the training phase, we employ the supervised learning and create a new labeled dataset for one-dimensional conservative approximation, where we construct a numerical flux function from the given point values such that the flux difference approximates the derivative to high-order accuracy. The symmetric-balancing term is introduced for the loss function so that it propels the neural network to match the conservative approximation to the derivative and satisfy the symmetric property that WENO3-JS and WENO3-Z have in common. The consequent WENO schemes, WENO3-CADNNs, demonstrate robust generalization across various benchmark scenarios and resolutions, where they outperform WENO3-Z and achieve accuracy comparable to WENO5-JS.