Nonlinear Model Reduction for Slow–Fast Stochastic Systems Near Unknown Invariant Manifolds

Abstract

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.