mlsanityData to 7 Oct 2026

Learning WENO for entropy stable schemes to solve conservation laws

arXiv 2403.14848 ↗First posted 21 Mar 2024math.NA cs.LG cs.NAcited by 1

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Abstract (arXiv)

Entropy conditions play a crucial role in the extraction of a physically relevant solution for systems of conservation laws, thus motivating the construction of entropy stable schemes that satisfy a discrete analogue of such conditions. TeCNO schemes (Fjordholm et al. 2012) form a class of arbitrary high-order entropy stable finite difference solvers, which require specialized reconstruction algorithms satisfying the sign property at each cell interface. Third-order weighted essentially non-oscillatory (WENO) schemes called SP-WENO (Fjordholm and Ray, 2016) and SP-WENOc (Ray, 2018) have been designed to satisfy the sign property. However, these WENO algorithms can perform poorly near shocks, with the numerical solutions exhibiting large spurious oscillations. In the present work, we propose a variant of the SP-WENO, termed as Deep Sign-Preserving WENO (DSP-WENO), where a neural network is trained to learn the WENO weighting strategy. The sign property and third-order accuracy are strongly imposed in the algorithm, which constrains the WENO weight selection region to a convex polygon. Thereafter, a neural network is trained to select the WENO weights from this convex region with the goal of improving the shock-capturing capabilities without sacrificing the rate of convergence in smooth regions. The proposed synergistic approach retains the mathematical framework of the TeCNO scheme while integrating deep learning to remedy the computational issues of the WENO-based reconstruction. We present several numerical experiments to demonstrate the significant improvement with DSP-WENO over the existing variants of WENO satisfying the sign property.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 18 of its 66 references that are in the map. First posted on arXiv 21 Mar 2024.

Builds on

18 of 66 references.

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6 more
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Citing papers per year: 2024: 1.

1 paper.

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