  We present a class of efficient parametric closure models for 1D stochastic
Burgers equations. Casting it as statistical learning of the flow map, we
derive the parametric form by representing the unresolved high wavenumber
Fourier modes as functionals of the resolved variables' trajectory. The reduced
models are nonlinear autoregression (NAR) time series models, with coefficients
estimated from data by least squares. The NAR models can accurately reproduce
the energy spectrum, the invariant densities, and the autocorrelations. Taking
advantage of the simplicity of the NAR models, we investigate maximal and
optimal space-time reduction. Reduction in space dimension is unlimited, and
NAR models with two Fourier modes can perform well. The NAR model's stability
limits time reduction, with a maximal time step smaller than that of the K-mode
Galerkin system. We report a potential criterion for optimal space-time
reduction: the NAR models achieve minimal relative error in the energy spectrum
at the time step where the K-mode Galerkin system's mean CFL number agrees with
the full model's.
