A learned closure method applied to phase mixing in a turbulent gradient-driven gyrokinetic system in simple geometry

We present a new method for formulating closures that learn from kinetic simulation data. We apply this method to phase mixing in a simple gyrokinetic turbulent system - temperature-gradient-driven turbulence in an unsheared slab. The closure, called the learned multi-mode (LMM) closure, is constructed by, first, extracting an optimal basis from a nonlinear kinetic simulation using singular value decomposition. Subsequent nonlinear fluid simulations are projected onto this basis and the results are used to formulate the closure. We compare the closure with other closures schemes over a broad range of the relevant two-dimensional parameter space (collisionality and gradient drive). We find that the turbulent kinetic system produces phase-mixing rates much lower than the linear expectations, which the LMM closure is capable of capturing. We also compare radial heat fluxes. A Hammett-Perkins closure, generalized to include collisional effects, is quite successful throughout the parameter space, producing ${∼ }14 %$ root-mean-square (r.m.s.) error. The LMM closure is also very effective: when trained at three (two) points (in a 35 point parameter grid), the LMM closure produces $8 %$ ($12 %$) r.m.s. errors. The LMM procedure can be readily generalized to other closure problems.