  We introduce a machine-learning framework named statistics-informed neural
network (SINN) for learning stochastic dynamics from data. This new
architecture was theoretically inspired by a universal approximation theorem
for stochastic systems, which we introduce in this paper, and the
projection-operator formalism for stochastic modeling. We devise mechanisms for
training the neural network model to reproduce the correct \emph{statistical}
behavior of a target stochastic process. Numerical simulation results
demonstrate that a well-trained SINN can reliably approximate both Markovian
and non-Markovian stochastic dynamics. We demonstrate the applicability of SINN
to coarse-graining problems and the modeling of transition dynamics.
Furthermore, we show that the obtained reduced-order model can be trained on
temporally coarse-grained data and hence is well suited for rare-event
simulations.
