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

Abstract Inspired by the gradient flow viewpoint of the Landau equation and the corresponding dynamic formulation of the Landau metric in Carrillo et al. (2024a), we develop a novel implicit particle method for the Landau equation in the framework of the Jordan–Kinderlehrer–Otto 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 have designed 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, thereby making it suitable for large-scale plasma simulations over extended time periods.