Learning WENO for entropy stable schemes to solve conservation laws

2024-03-21

arXiv.org

AI Summary

Conventional SP-WENO and SP-WENOc schemes suffer from spurious oscillations near shocks and struggle to simultaneously ensure entropy stability and sign preservation. Method: This paper proposes Deep Sign-Preserving WENO (DSP-WENO), the first WENO scheme embedding a deep neural network directly into the nonlinear weight selection process. Within the TeCNO entropy-stable framework, DSP-WENO explicitly enforces sign constraints and performs optimization over a convex polygonal feasible domain, rigorously preserving third-order accuracy and mathematical consistency. Results: DSP-WENO substantially suppresses non-physical oscillations across strong discontinuities while retaining optimal convergence rates in smooth regions. Numerical validation across conservation laws—including the Euler equations—demonstrates superior accuracy and robustness compared to SP-WENO and SP-WENOc. The method establishes a new paradigm for high-resolution shock-capturing that unifies theoretical guarantees with data-driven adaptivity.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

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• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
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Abstract

Entropy conditions play a crucial role in the extraction of a physically relevant solution for a system 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. Recently, third-order 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.

Problem

• Improving WENO schemes for entropy stable solutions of conservation laws
• Reducing spurious oscillations in WENO near shocks while preserving accuracy
• Integrating neural networks to enhance WENO weight selection for shock-capturing

Innovation

• Neural network learns WENO weighting strategy
• Constrains WENO weights to convex polygon
• Improves shock-capturing without sacrificing accuracy