Semi-Lagrangian (SL) schemes are highly efficient for simulating transport equations and are widely used across various applications. Despite their success, designing genuinely multi-dimensional and conservative SL schemes remains a significant challenge. Building on our previous work [Chen et al., J. Comput. Phys., V490 112329, (2023)], we introduce a conservative machine-learning-based SL finite difference (FD) method that allows for extra-large time step evolution. At the core of our approach is a novel dynamical graph neural network designed to handle the complexities associated with tracking accurately upstream points along characteristics. This proposed neural transport solver learns the conservative SL FD discretization directly from data, improving accuracy and efficiency compared to traditional numerical schemes, while significantly simplifying algorithm implementation. We validate the method' s effectiveness and efficiency through numerical tests on benchmark transport equations in both one and two dimensions, as well as the nonlinear Vlasov-Poisson system.

翻译：半拉格朗日（SL）格式在输运方程模拟中具有极高的效率，并广泛应用于各类场景。尽管取得了成功，但设计真正多维且守恒的SL格式仍是一项重大挑战。基于我们先前的工作[Chen et al., J. Comput. Phys., V490 112329, (2023)]，本文提出了一种基于机器学习的保守型SL有限差分（FD）方法，该算法支持超大时间步长演化。我们方法的核心是一种新型动态图神经网络，旨在处理沿特征线精确追踪上游点所涉及的复杂性。该神经输运求解器直接从数据中学习守恒型SL-FD离散形式，相较于传统数值格式，在提高精度和效率的同时显著简化了算法实现。通过一维和二维基准输运方程以及非线性Vlasov-Poisson系统的数值测试，我们验证了该方法的有效性和效率。