Conservative approximation-based feedforward neural network for WENO schemes

• URL: http://arxiv.org/abs/2507.06190v1
• Date: Tue, 08 Jul 2025 17:19:48 GMT
• Title: Conservative approximation-based feedforward neural network for WENO schemes
• Authors: Kwanghyuk Park, Jiaxi Gu, Jae-Hun Jung,
• Abstract summary: We present the feedforward neural network based on the conservative approximation to the derivative from point values.<n>We present WENO3-CADNNs, where they outperform WENO3-Z and achieve accuracy comparable to WENO5-JS.
• Score: 4.867849275247251
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

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