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.

翻译：熵条件在提取守恒律系统的物理相关解中起着关键作用，这促使了满足离散熵条件的熵稳定格式的构造。TeCNO格式（Fjordholm等，2012）是一类任意高阶的熵稳定有限差分解算器，其要求在每个单元界面满足符号性质的特殊重构算法。近期，三阶WENO格式SP-WENO（Fjordholm和Ray，2016）及SP-WENOc（Ray，2018）被设计为满足该符号性质，然而这些WENO算法在激波附近表现欠佳，数值解会出现大幅伪振荡。本文提出SP-WENO的变体——深度保符号WENO（DSP-WENO），通过训练神经网络学习WENO权重策略。该算法强约束符号性质和三阶精度，将WENO权重选择区域限制为凸多边形，随后训练神经网络从该凸区域中选取权重，旨在不牺牲光滑区域收敛速度的前提下提升激波捕捉能力。这种协同方法在保留TeCNO格式数学框架的同时，融合深度学习解决了基于WENO重构的计算问题。数值实验表明，与现有满足符号性质的WENO变体相比，DSP-WENO展现出显著改进。