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We introduce a nonlinear stochastic model reduction technique forhigh-dimensional stochastic dynamical systems that have a low-dimensionalinvariant effective manifold with slow dynamics, and high-dimensional, largefast modes. Given only access to a black box simulator from which short burstsof simulation can be obtained, we design an algorithm that outputs an estimateof the invariant manifold, a process of the effective stochastic dynamics onit, which has averaged out the fast modes, and a simulator thereof. Thissimulator is efficient in that it exploits of the low dimension of theinvariant manifold, and takes time steps of size dependent on the regularity ofthe effective process, and therefore typically much larger than that of theoriginal simulator, which had to resolve the fast modes. The algorithm and theestimation can be performed on-the-fly, leading to efficient exploration of theeffective state space, without losing consistency with the underlying dynamics.This construction enables fast and efficient simulation of paths of theeffective dynamics, together with estimation of crucial features andobservables of such dynamics, including the stationary distribution,identification of metastable states, and residence times and transition ratesbetween them.

Nonlinear model reduction for slow-fast stochastic systems near unknown invariant manifolds