Learning stochastic multiscale models through normalizing flows

2026-05-10

AI Summary

This work addresses the challenge of learning stochastic multiscale systems with unobserved fast processes from a single trajectory of slow variables, where the invariant distribution of the fast dynamics is unknown. The authors propose an end-to-end learning framework based on stochastic differential equations that integrates stochastic averaging for structure-preserving dimensionality reduction. Crucially, they introduce normalizing flows to flexibly parameterize the invariant distribution of the latent fast variables—a first in this context—and combine this with variational Bayesian inference to quantify epistemic uncertainty in model parameters. Using only a single observed slow-variable trajectory, the method accurately identifies the effective stochastic dynamics, achieving both scalability and significantly enhanced modeling robustness and reliability of uncertainty quantification.

Abstract

Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a single trajectory of the slow component, while the fast dynamics remain unobserved, making statistical learning challenging. Approaches based on partial differential equations (PDE), such as Fokker-Planck formulations, aim to characterize the evolution of probability densities, typically requiring dense space-time data or grid-based solvers. In contrast, we adopt a trajectory-based perspective and develop a data-driven framework for learning effective stochastic dynamics from a single observed path. We model the dynamics by coupled multiscale stochastic differential equations (SDEs) and first obtain a principled model reduction through stochastic averaging. Unlike generic model reduction techniques such as PCA, this respects the dynamical structure of the original system and explicitly incorporates the interaction between slow and fast scales. A central challenge, however, is that the reduced model depends on the invariant distribution of the fast process, which is a solution to an intractable and often unknown PDE. We introduce a novel learning framework that parameterizes the invariant distribution using normalizing flows, enabling expressive density modeling in the latent fast-variable space. The flow is trained end-to-end by optimizing a penalized likelihood objective induced by the reduced stochastic dynamics. Furthermore, we develop a Bayesian variational inference procedure for uncertainty quantification, employing a second normalizing flow to approximate the posterior distribution over model parameters. This yields a scalable approach to capturing epistemic uncertainty in multiscale systems.

Problem

• multiscale stochastic dynamics
• model reduction
• invariant distribution
• data-driven learning
• unobserved fast variables

Innovation

• normalizing flows
• multiscale stochastic dynamics
• stochastic averaging
• invariant distribution learning
• Bayesian variational inference