Auto-SDE: Learning effective reduced dynamics from data-driven stochastic dynamical systems

Multiscale stochastic dynamical systems have been widely adopted to scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective reduced dynamics for a slow-fast stochastic dynamical system. Given observation data on a short-term period satisfying some unknown slow-fast stochastic system, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also proved to be accurate, stable and effective through numerical experiments under various evaluation metrics.