Generative Modeling of Stochastic Dynamics for Long-Time Evolution

2026-09-30

AI Summary

This study addresses the challenge of predicting long-term evolution in non-equilibrium stochastic dynamics, where exact governing equations are typically unknown. We propose an equation-free, data-driven framework that learns finite-time transition kernels from configuration pairs separated by a fixed short time step using generative diffusion models, enabling long-term dynamical predictions through iterative propagation. This approach overcomes the reliance on known equations of motion, allowing reconstruction and extrapolation of long-time non-equilibrium behavior solely from short-time observational data. We successfully reproduce dynamic critical scaling and self-similar coarsening in two-dimensional Model B, and accurately predict particle currents and mean first-passage times in experimental colloidal systems.

Abstract

Exact stochastic equations for non-equilibrium dynamics are rarely accessible. We show that the long-time evolution of stochastic dynamics can be predicted from configuration pairs at a fixed short time lag, without knowledge of the equation of motion. Generative diffusion models learn the finite-time transition kernel from these pairs, and iterating it propagates the dynamics far beyond the training lag. For two-dimensional Model B, the diffusive dynamics of a conserved order parameter, the learned kernels reproduce dynamic critical scaling and self-similar $t^{1/3}$ coarsening. Agreement with direct simulations persists on lattices twice the largest training size and for initial ensembles absent from training. For driven colloids in a periodic optical potential, ten minutes of measured trajectories suffice to predict the particle current and mean passage time over the next twenty minutes within experimental uncertainty. Short-time observations thus contain the information needed to predict emergent non-equilibrium dynamics at much longer times.

Problem

• stochastic dynamics
• non-equilibrium dynamics
• long-time evolution
• generative modeling

Innovation

• Generative diffusion models
• Stochastic dynamics
• Transition kernel
• Non-equilibrium dynamics
• Long-time evolution