A neural operator closure for Landau damping in electrostatic plasma

We present a data-driven plasma fluid closure for both linear and nonlinear electrostatic Landau damping in one dimension. A Fourier neural operator (FNO) is trained online within a differentiable fluid solver, with the loss computed on trajectories produced by the closed fluid simulation rather than on individual kinetic snapshots. The closure is non-Markovian, acting on a trailing window of the resolved-moment history so as to represent the memory of the unresolved dynamics. We demonstrate that a single FNO trained in this way reproduces both linear and nonlinear Landau damping, generalises to unseen initial perturbation amplitudes lying between those used in training, and remains numerically stable when deployed in independent fluid simulations. In the nonlinear regime the learned heat flux reproduces the resolved-moment dynamics without matching the kinetic heat flux pointwise, behaving as an effective closure that carries the influence of the unresolved dynamics through the resolved-field history, though the learned specific flux is expected to depend on the numerical scheme and training data. A sensitivity analysis of the trained model shows that it computes a genuine moment-to-flux relation whose reliance on the memory window is structured rather than uniform, concentrating at lags spaced by the oscillation period.