We identify effective stochastic differential equations (SDE) for coarse
observables of fine-grained particle- or agent-based simulations; these SDE
then provide 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 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. 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.Comment: 19 pages, includes supplemental materia