  We propose a high-order stochastic-statistical moment closure model for
efficient ensemble prediction of leading-order statistical moments and
probability density functions in multiscale complex turbulent systems. The
statistical moment equations are closed by a precise calibration of the
high-order feedbacks using ensemble solutions of the consistent stochastic
equations, suitable for modeling complex phenomena including non-Gaussian
statistics and extreme events. To address challenges associated with closely
coupled spatio-temporal scales in turbulent states and expensive large ensemble
simulation for high-dimensional systems, we introduce efficient computational
strategies using the random batch method (RBM). This approach significantly
reduces the required ensemble size while accurately capturing essential
high-order structures. Only a small batch of small-scale fluctuation modes is
used for each time update of the samples, and exact convergence to the full
model statistics is ensured through frequent resampling of the batches during
time evolution. Furthermore, we develop a reduced-order model to handle systems
with really high dimension by linking the large number of small-scale
fluctuation modes to ensemble samples of dominant leading modes. The
effectiveness of the proposed models is validated by numerical experiments on
the one-layer and two-layer Lorenz '96 systems, which exhibit representative
chaotic features and various statistical regimes. The full and reduced-order
RBM models demonstrate uniformly high skill in capturing the time evolution of
crucial leading-order statistics, non-Gaussian probability distributions, while
achieving significantly lower computational cost compared to direct Monte-Carlo
approaches.
