  A new ensemble forecast algorithm, named as the physics-informed data-driven
algorithm with conditional Gaussian statistics (PIDD-CG), is developed to
predict the time evolution of the probability density functions (PDFs) of
complex turbulent systems with partial observations. The PIDD-CG algorithm
integrates a unique multiscale statistical closure model with an extremely
efficient nonlinear data assimilation scheme to represent the PDF as a mixture
of conditional statistics, which overcomes the curse of dimensionality for
high-dimensional systems. The multiscale features in the time evolution of each
conditional statistics ensemble member effectively captured by an appropriate
combination of physics-informed analytic formulae and recurrent neural
networks. An information metric is adopted as the loss function for the latter
to more accurately calibrate the key turbulent signals with strong
fluctuations. The proposed algorithm succeeds in forecasting both the transient
and statistical equilibrium non-Gaussian PDFs of strongly turbulent systems
with intermittency, regime switching and extreme events.
