  We present a numerical method for learning unknown nonautonomous stochastic
dynamical system, i.e., stochastic system subject to time dependent excitation
or control signals. Our basic assumption is that the governing equations for
the stochastic system are unavailable. However, short bursts of input/output
(I/O) data consisting of certain known excitation signals and their
corresponding system responses are available. When a sufficient amount of such
I/O data are available, our method is capable of learning the unknown dynamics
and producing an accurate predictive model for the stochastic responses of the
system subject to arbitrary excitation signals not in the training data. Our
method has two key components: (1) a local approximation of the training I/O
data to transfer the learning into a parameterized form; and (2) a generative
model to approximate the underlying unknown stochastic flow map in
distribution. After presenting the method in detail, we present a comprehensive
set of numerical examples to demonstrate the performance of the proposed
method, especially for long-term system predictions.
