  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 design an algorithm that outputs an estimate
of the invariant manifold, a process of the effective stochastic dynamics on
it, which has averaged out the fast modes, and a simulator thereof. This
simulator is efficient in that it exploits of the low dimension of the
invariant manifold, and takes time steps of size dependent on the regularity of
the effective process, and therefore typically much larger than that of the
original simulator, which had to resolve the fast modes. The algorithm and the
estimation 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.
