标题： 一种数据驱动的有限体积格式中双曲守恒定律和变化边界条件的学习离散化方法 显示英文标题

标题： A data-driven learned discretization approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions

摘要： 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偏微分方程。 这项工作建立在之前的研究基础上，该研究结合了对标量守恒定律平滑解的数据驱动有限差分近似，其中通过这些方程的准确但繁琐的解来学习近似空间导数的神经网络最优系数。 我们将这种方法扩展到适用于双曲标量和守恒定律系统的通量限制有限体积格式。 我们还训练离散化方法，以高效捕捉具有激波和接触波的不连续解，并应用于边界条件。 通过定义新的损失函数、填充和充分的数据集，扩展了数据驱动模型的学习过程。 这些新元素保证了计算稳定性，保持了细网格解的准确性，并提高了整体性能。 使用文献中的一维和二维测试案例进行的数值实验表明，所学模型能够在非常粗略的网格上准确再现细网格结果。 显示英文摘要

摘要： 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.