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Lagrangian(SL)计划被称为解决运输方程的主要数字工具,具有许多优势,并被广泛用于计算流体动态、等离子物理模型、数字天气预测等领域,在这项工作中,我们开发了新型机器辅助学习方法,以加速传统的SL有限体积(FV)计划;拟议计划避免了对上游细胞进行昂贵的跟踪,但试图通过将特定的感应偏差纳入神经网络,大大简化算法的实施并导致效率的提高,从数据中学习SL分解;此外,该方法提供了剧烈的震荡过渡和精确度,通常需要与传统运输求解器建立更细的电网格;数字测试显示了拟议方法的有效性和效率。