Machine-learning Closure for Vlasov-Poisson Dynamics in Fourier-Hermite Space

TL;DR

This work uses reservoir computing, a machine-learning (ML) architecture, to develop a closure model for a limiting case of electrostatic gyrokinetics and exploits the locality of interactions in the Hermite representation to introduce an ML closure model of the small-scale dynamics in velocity space.

Abstract

Accurate reduced models of turbulence are desirable to facilitate the optimization of magnetic-confinement fusion reactor designs. As a first step toward higher-dimensional turbulence applications, we use reservoir computing, a machine-learning (ML) architecture, to develop a closure model for a limiting case of electrostatic gyrokinetics. We implement a pseudo-spectral Eulerian code to solve the one-dimensional Vlasov-Poisson system on a basis of Fourier modes in configuration space and Hermite polynomials in velocity space. When cast onto the Hermite basis, the Vlasov equation becomes an infinitely coupled hierarchy of fluid moments, presenting a closure problem. We exploit the locality of interactions in the Hermite representation to introduce an ML closure model of the small-scale dynamics in velocity space. In the linear limit, when the kinetic Fourier-Hermite solver is augmented with the reservoir closure, the closure permits a reduction of the velocity resolution, with a relative error within two percent for the Hermite moment where the reservoir closes the hierarchy. In the strongly-nonlinear regime, the ML closure model more accurately resolves the low-order Fourier and Hermite spectra when compared to a naïve closure by truncation and reduces the required velocity resolution by a factor of sixteen.

Figures (15)

• Figure 1: Diagram of the reservoir computing closure model, integrated with the kinetic moment solver.
• Figure 2: High-velocity-resolution ($M=1025$) baseline time series of Hermite amplitudes for the driven, linearized Vlasov-Poisson system. Energy is injected as a density perturbation in the $m=0$ Hermite moment and advects to higher moments through linear Landau damping. No hypercollisional regularization is applied to this example, and hundreds of Hermite moments are required to prevent numerical reflection at the high-$m$ boundary from impacting the low-$m$ spectrum.
• Figure 3: Time-averaged Hermite spectra for the ML closure model compared to the high-velocity-resolution ($M=1025$) baseline, closure-by-truncation with three different coefficients ($\nu_m$) for hypercollisional regularization, and the theoretical $m^{-1/2}$ scaling. The ML closure model shows strong agreement with the baseline simulation, while no value of $\nu_m$ produces an accurate spectrum.
• Figure 4: Comparison between baseline numerical solution, ML closure, and theoretical damping rate of the Fourier-Hermite amplitude for an initial cosine density perturbation. The low-amplitude perturbation shows strong agreement with the theoretical damping rate. When augmented with the ML closure, the moment solver continues to capture the behavior well at a lower Hermite resolution of $M=4$, as opposed to the $M=17$ baseline.
• Figure 5: Hermite spectra of the high-velocity-resolution ($M=17$), truncated ($M=4$) simulations, and ML closure for the initial value problem in Figure \ref{['fig:linear_landau']} . The spectra are averaged over time and Fourier wavenumber. The ML closure model permits a low-resolution simulation to accurately resolve the Hermite spectrum.