Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning
[1];
[4];
[4]
- Department of Informatics, School of Computation, Information and Technology, Technical University of Munich, 80333 Munich, Germany
- Faculty of Computational Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia
- Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Department of Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA
- Institute of Mathematics, University of Potsdam, 14469 Potsdam, Germany
We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs 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 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.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 1924816
- Journal Information:
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Journal Name: Chaos: An Interdisciplinary Journal of Nonlinear Science Vol. 33 Journal Issue: 2; ISSN 1054-1500
- Publisher:
- American Institute of PhysicsCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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