Semi-Lagrangian (SL) schemes are known as a major numerical tool for solving transport equations with many advantages and have been widely deployed in the fields of computational fluid dynamics, plasma physics modeling, numerical weather prediction, among others. In this work, we develop a novel machine learning-assisted approach to accelerate the conventional SL finite volume (FV) schemes. The proposed scheme avoids the expensive tracking of upstream cells but attempts to learn the SL discretization from the data by incorporating specific inductive biases in the neural network, significantly simplifying the algorithm implementation and leading to improved efficiency. In addition, the method delivers sharp shock transitions and a level of accuracy that would typically require a much finer grid with traditional transport solvers. Numerical tests demonstrate the effectiveness and efficiency of the proposed method.

翻译：半拉格朗日（SL）格式是求解传输方程的重要数值工具，具有诸多优势，已广泛应用于计算流体动力学、等离子体物理建模、数值天气预报等领域。本研究提出一种新颖的机器学习辅助方法，用于加速传统半拉格朗日有限体积（FV）格式。所提出的格式避免了昂贵的上游单元追踪，而是通过在神经网络中融入特定归纳偏置，从数据中学习半拉格朗日离散化，从而显著简化算法实现并提高效率。此外，该方法能够呈现尖锐的激波过渡，并达到传统传输求解器在更细网格下才能实现的精度水平。数值实验验证了所提方法的有效性和效率。