Machine Learning Heat Flux Closure for Multi-Moment Fluid Modeling of Nonlinear Landau Damping*

Nonlinear plasma physics problems are typically simulated through comprehensive modeling of phase space, incurring extreme computational costs. This has driven the development of multi-moment fluid models that integrates the Vlasov equation. However, a significant challenge remains in identifying a suitable fluid closure for these models. Recent advancements in physics-informed machine learning have sparked renewed interests in constructing accurate fluid closure terms. In this study, we present a novel approach that integrates kinetic physics from first principal Vlasov simulation data into a multi-moment fluid model through the heat flux closure term using the Fourier neural operator (FNO) —a specialized neural network architecture. For the first time, without resolving phase space dynamics, the newly developed fluid model accurately captures the nonlinear evolution of the Landau damping process, matching the fully kinetic simulation data precisely. This machine learning-assisted framework provides a computationally affordable method that surpasses previous fluid models in accurately modeling the kinetic evolution of complex plasma systems.