  We present a numerical framework for learning unknown stochastic dynamical
systems using measurement data. Termed stochastic flow map learning (sFML), the
new framework is an extension of flow map learning (FML) that was developed for
learning deterministic dynamical systems. For learning stochastic systems, we
define a stochastic flow map that is a superposition of two sub-flow maps: a
deterministic sub-map and a stochastic sub-map. The stochastic training data
are used to construct the deterministic sub-map first, followed by the
stochastic sub-map. The deterministic sub-map takes the form of residual
network (ResNet), similar to the work of FML for deterministic systems. For the
stochastic sub-map, we employ a generative model, particularly generative
adversarial networks (GANs) in this paper. The final constructed stochastic
flow map then defines a stochastic evolution model that is a weak
approximation, in term of distribution, of the unknown stochastic system. A
comprehensive set of numerical examples are presented to demonstrate the
flexibility and effectiveness of the proposed sFML method for various types of
stochastic systems.
