标题： 学习求解双曲方程的高精度粗网格数值格式 显示英文标题

标题： Learning high-accuracy numerical schemes for hyperbolic equations on coarse meshes

摘要： 利用经典的有限差分法或有限体积法求解偏微分方程时，需要足够精细的网格以及精心设计的格式才能获得数值解的高阶精度，这会导致计算成本显著增加，特别是在三维时间相关的场景中。最近，提出了结合机器学习辅助的数值方法来提高准确性和效率。本文提出了一种基于数据驱动的有限差分数值方法，用于在粗网格上求解具有光滑解的双曲型方程，该方法能够比基于相同网格尺寸的经典数值格式实现更高的精度。此外，尽管训练过程中未显式优化频谱特性，但数据驱动格式的频谱特性优于经典格式。数值实验展示了所提出方法的准确性和效率，以及其在色散和耗散方面的良好性能。 显示英文摘要

摘要： When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical solutions, leading to a significant increase in computational costs, especially for three-dimensional (3D) time-dependent problems. Recently, machine learning-assisted numerical methods have been proposed to enhance accuracy or efficiency. In this paper, we propose a data-driven finite difference numerical method to solve the hyperbolic equations with smooth solutions on coarse grids, which can achieve higher accuracy than classical numerical schemes based on the same mesh size. In addition, the data-driven schemes have better spectrum properties than the classical schemes, although the spectrum properties are not explicitly optimized during the training process. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method, as well as its good performance on dispersion and dissipation.