用数据驱动方法优化有限体积法，实现粗网格下高精度求解双曲守恒律。
A data-driven learned discretization approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions
- 通过神经网络学习最优离散化系数，替代传统数值格式。
- 在粗网格上复现细网格的精确解，误差极小且稳定。
- 适用于带间断波和变化边界的复杂问题，适合科学计算领域研究者。
本文提出一种数据驱动的有限体积方法，用于求解一维和二维双曲型偏微分方程。该工作扩展了先前基于神经网络学习光滑解空间导数最优系数的数据驱动有限差分方法，将其推广至通量限制的有限体积格式，适用于标量及系统守恒律。模型通过新定义的损失函数、填充策略与充分数据集训练，能高效捕捉含激波与接触间断的不连续解，并支持变化边界条件。数值实验在文献标准测试案例中验证：即使在极粗网格上，所学模型仍能准确复现细网格解，保持计算稳定性并提升整体性能。
原文摘要 · Abstract (English)
This paper presents a data-driven finite volume method for solving 1D and 2D hyperbolic partial differential equations. This work builds upon the prior research incorporating a data-driven finite-difference approximation of smooth solutions of scalar conservation laws, where optimal coefficients of neural networks approximating space derivatives are learned based on accurate, but cumbersome solutions to these equations. We extend this approach to flux-limited finite volume schemes for hyperbolic scalar and systems of conservation laws. We also train the discretization to efficiently capture discontinuous solutions with shock and contact waves, as well as to the application of boundary conditions. The learning procedure of the data-driven model is extended through the definition of a new loss, paddings and adequate database. These new ingredients guarantee computational stability, preserve the accuracy of fine-grid solutions, and enhance overall performance. Numerical experiments using test cases from the literature in both one- and two-dimensional spaces demonstrate that the learned model accurately reproduces fine-grid results on very coarse meshes.
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