  We formulate a reduced-order strategy for efficiently forecasting complex
high-dimensional dynamical systems entirely based on data streams. The first
step of our method involves reconstructing the dynamics in a reduced-order
subspace of choice using Gaussian Process Regression (GPR). GPR simultaneously
allows for reconstruction of the vector field and more importantly, estimation
of local uncertainty. The latter is due to i) local interpolation error and ii)
truncation of the high-dimensional phase space. This uncertainty component can
be analytically quantified in terms of the GPR hyperparameters. In the second
step we formulate stochastic models that explicitly take into account the
reconstructed dynamics and their uncertainty. For regions of the attractor
which are not sufficiently sampled for our GPR framework to be effective, an
adaptive blended scheme is formulated to enforce correct statistical steady
state properties, matching those of the real data. We examine the effectiveness
of the proposed method to complex systems including the Lorenz 96, the
Kuramoto-Sivashinsky, as well as a prototype climate model. We also study the
performance of the proposed approach as the intrinsic dimensionality of the
system attractor increases in highly turbulent regimes.
