A score-based particle method for homogeneous Landau equation

TL;DR

This paper introduces a novel score-based particle method for solving the Landau equation in plasmas, combining learning with structure-preserving techniques to improve scalability, efficiency, and theoretical understanding.

Contribution

The paper presents a new particle method that uses score-matching to handle nonlinearity, avoiding kernel density estimation and enabling better scalability and theoretical control.

Findings

Method effectively controls KL divergence with score-matching loss

Achieves conservation properties of deterministic particle methods

Demonstrates efficiency on Coulomb interaction example

Abstract

We propose a novel score-based particle method for solving the Landau equation in plasmas, that seamlessly integrates learning with structure-preserving particle methods [arXiv:1910.03080]. Building upon the Lagrangian viewpoint of the Landau equation, a central challenge stems from the nonlinear dependence of the velocity field on the density. Our primary innovation lies in recognizing that this nonlinearity is in the form of the score function, which can be approximated dynamically via techniques from score-matching. The resulting method inherits the conservation properties of the deterministic particle method while sidestepping the necessity for kernel density estimation in [arXiv:1910.03080]. This streamlines computation and enhances scalability with dimensionality. Furthermore, we provide a theoretical estimate by demonstrating that the KL divergence between our approximation and the true solution can be effectively controlled by the score-matching loss. Additionally, by adopting the flow map viewpoint, we derive an update formula for exact density computation. Extensive examples have been provided to show the efficiency of the method, including a physically relevant case of Coulomb interaction.