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We identify effective stochastic differential equations (SDE) for coarseobservables of fine-grained particle- or agent-based simulations; these SDEthen provide useful coarse surrogate models of the fine scale dynamics. Weapproximate the drift and diffusivity functions in these effective SDE throughneural networks, which can be thought of as effective stochastic ResNets. Theloss function is inspired by, and embodies, the structure of establishedstochastic numerical integrators (here, Euler-Maruyama and Milstein); ourapproximations can thus benefit from backward error analysis of theseunderlying numerical schemes. They also lend themselves naturally to"physics-informed" gray-box identification when approximate coarse models, suchas mean field equations, are available. Existing numerical integration schemesfor Langevin-type equations and for stochastic partial differential equations(SPDE) can also be used for training; we demonstrate this on a stochasticallyforced oscillator and the stochastic wave equation. Our approach does notrequire long trajectories, works on scattered snapshot data, and is designed tonaturally handle different time steps per snapshot. We consider both the casewhere the coarse collective observables are known in advance, as well as thecase where they must be found in a data-driven manner.

Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning