JKO for Landau: a variational particle method for homogeneous Landau equation

TL;DR

This paper introduces a novel implicit particle method based on the JKO scheme for the Landau equation, utilizing neural networks and stochastic gradient descent to achieve stability and efficiency in plasma simulations.

Contribution

The authors develop a new variational particle method for the Landau equation that incorporates neural networks and stochastic optimization, ensuring entropy dissipation and stability.

Findings

Method achieves exact entropy dissipation.

Significantly reduces computational complexity.

Suitable for large-scale plasma simulations.

Abstract

Inspired by the gradient flow viewpoint of the Landau equation and the corresponding dynamic formulation of the Landau metric in [arXiv:2007.08591], we develop a novel implicit particle method for the Landau equation in the framework of the JKO scheme. We first reformulate the Landau metric in a computationally friendly form, and then translate it into the Lagrangian viewpoint using the flow map. A key observation is that, while the flow map evolves according to a rather complicated integral equation, the unknown component is simply a score function of the corresponding density plus an additional term in the null space of the collision kernel. This insight guides us in designing and training the neural network for the flow map. Additionally, the objective function is in a double summation form, making it highly suitable for stochastic methods. Consequently, we design a tailored version of stochastic gradient descent that maintains particle interactions and significantly reduces the computational complexity. Compared to other deterministic particle methods, the proposed method enjoys exact entropy dissipation and unconditional stability, therefore making it suitable for large-scale plasma simulations over extended time periods.