  We identify effective stochastic differential equations (SDE) for coarse
observables of fine-grained particle- or agent-based simulations; these SDE
then provide useful coarse surrogate models of the fine scale dynamics. We
approximate the drift and diffusivity functions in these effective SDE through
neural networks, which can be thought of as effective stochastic ResNets. The
loss function is inspired by, and embodies, the structure of established
stochastic numerical integrators (here, Euler-Maruyama and Milstein); our
approximations can thus benefit from backward error analysis of these
underlying numerical schemes. They also lend themselves naturally to
"physics-informed" gray-box identification when approximate coarse models, such
as mean field equations, are available. Existing numerical integration schemes
for Langevin-type equations and for stochastic partial differential equations
(SPDE) can also be used for training; we demonstrate this on a stochastically
forced oscillator and the stochastic wave equation. Our approach does not
require long trajectories, works on scattered snapshot data, and is designed to
naturally handle different time steps per snapshot. We consider both the case
where the coarse collective observables are known in advance, as well as the
case where they must be found in a data-driven manner.
