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.

翻译：本文提出了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。本研究基于先前将数据驱动的有限差分近似应用于标量守恒律光滑解的工作，其中逼近空间导数的神经网络最优系数是通过对这些方程的精确但繁琐的解进行学习而获得的。我们将此方法推广至适用于双曲标量及守恒律方程组的通量限制型有限体积格式。我们还训练该离散化方法以有效捕捉含激波和接触间断的不连续解，并将其应用于边界条件的处理。通过定义新的损失函数、填充策略及适当数据库，扩展了数据驱动模型的学习过程。这些新要素保证了计算稳定性，保持了细网格解的精度，并提升了整体性能。利用文献中的一维和二维测试案例进行的数值实验表明，学习后的模型能够在极粗网格上精确复现细网格结果。