FPL-net: A deep learning framework for solving the nonlinear Fokker–Planck–Landau collision operator for anisotropic temperature relaxation

Abstract

The nonlinear collision operator consumes a significant amount of computation time in tokamak whole-volume modeling, and in current numerical methods, the computational time grows $O\left(n^2\right)$, with n representing the number of plasma species. In this study, we address the acceleration of the Fokker–Planck–Landau (FPL) collision operator using deep learning techniques. The developed FPL-net, a deep learning-based nonlinear Fokker–Planck–Landau collision operator, is a fully convolutional neural network optimized for computational speed with a compact model structure. FPL-net was trained on data representing various temperature conditions of an electron plasma on a two-dimensional velocity grid, ensuring generality. The network's training incorporated physics-informed loss functions for density, momentum, and energy moments of the plasma probability distribution function, which served as constraints, and it was trained to recursively predict two time steps, achieving robust accuracy. Notably, FPL-net demonstrated full temperature relaxation, representing the first time this has been accomplished by a deep learning–based FPL collision operator. Additional experiments with noisy inputs and extended rollouts validated the model's accuracy, which also shows over 1000x acceleration compared to traditional finite volume methods. We discuss the achieved acceleration through deep learning techniques and propose potential avenues for further enhancement and refinement in future research.