  We introduce a nonlinear stochastic model reduction technique for
high-dimensional stochastic dynamical systems that have a low-dimensional
invariant effective manifold with slow dynamics, and high-dimensional, large
fast modes. Given only access to a black box simulator from which short bursts
of simulation can be obtained, we estimate the invariant manifold, a process of
the effective (stochastic) dynamics on it, and construct an efficient simulator
thereof. These estimation steps can be performed on-the-fly, leading to
efficient exploration of the effective state space, without losing consistency
with the underlying dynamics. This construction enables fast and efficient
simulation of paths of the effective dynamics, together with estimation of
crucial features and observables of such dynamics, including the stationary
distribution, identification of metastable states, and residence times and
transition rates between them.
