  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 (SVD). 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 2D 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 (RMS) error. The LMM closure is also very
effective: when trained at three (two) points (in a 35 point parameter grid),
the LMM closure produces 9% (12%) RMS errors. The LMM procedure can be readily
generalized to other closure problems.

Plasma Physics