A learned conservative semi-Lagrangian finite volume scheme for transport simulations

TL;DR

This paper introduces a machine learning-enhanced semi-Lagrangian finite volume scheme that accelerates transport simulations, simplifies implementation, and maintains high accuracy with sharp shock resolution, outperforming traditional methods.

Contribution

It presents a novel neural network-based approach to learn semi-Lagrangian discretization, reducing computational cost and improving accuracy without tracking upstream cells.

Findings

Achieves sharper shock transitions compared to traditional schemes.

Demonstrates significant efficiency improvements in numerical tests.

Maintains high accuracy with coarser grids.

Abstract

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.