Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning
This provides a coarse surrogate modeling approach for fine-scale particle- or agent-based simulations, enabling efficient analysis and prediction in complex systems, though it is incremental as it builds on existing stochastic numerical integrators.
The authors developed a method to learn effective stochastic differential equations (SDEs) from microscopic simulations by approximating drift and diffusivity functions with neural networks, linking stochastic numerics to deep learning, and demonstrated it on examples like a stochastically forced oscillator and the stochastic wave equation without requiring long trajectories.
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.