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.

翻译：当使用有限差分或有限体积法等经典格式求解偏微分方程时，需要足够精细的网格和精心设计的格式才能实现数值解的高阶精度，这导致计算成本显著增加，特别是对于三维（3D）时间相关问题。近年来，人们提出了机器学习辅助的数值方法来提高精度或效率。本文提出了一种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方程，该方法能够基于相同的网格尺寸获得比经典数值格式更高的精度。此外，数据驱动格式比经典格式具有更好的频谱特性，尽管在训练过程中并未显式优化频谱特性。数值算例展示了所提方法的精度和效率，以及其在色散和耗散方面的良好性能。