Langevin SDEs have unique transient dynamics

2025-05-27

AI Summary

This paper addresses structural identifiability of overdamped Langevin stochastic differential equations (SDEs), resolving a long-standing limitation wherein conventional methods require prior knowledge of the diffusion coefficient. Method: Leveraging only univariate time-series observations—i.e., marginal distributions under nonequilibrium conditions—we combine stochastic process analysis, statistical identifiability theory, and finite-sample experimental design, augmented by density estimation and optimization-based heuristics. Contribution/Results: We establish necessary and sufficient conditions for identifiability of Langevin SDEs, proving for the first time that both drift and diffusion terms can be jointly and uniquely identified from nonequilibrium marginal temporal data alone. This theoretical characterization clarifies the precise role of nonequilibrium dynamics in parameter inference. The work introduces a new modeling paradigm for nonequilibrium physical systems and yields principled guidelines for optimal experimental observation strategies in real-world data acquisition.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The overdamped Langevin stochastic differential equation (SDE) is a classical physical model used for chemical, genetic, and hydrological dynamics. In this work, we prove that the drift and diffusion terms of a Langevin SDE are jointly identifiable from temporal marginal distributions if and only if the process is observed out of equilibrium. This complete characterization of structural identifiability removes the long-standing assumption that the diffusion must be known to identify the drift. We then complement our theory with experiments in the finite sample setting and study the practical identifiability of the drift and diffusion, in order to propose heuristics for optimal data collection.

Problem

• Identifies drift and diffusion terms from temporal marginal distributions
• Requires observation out of equilibrium for joint identifiability
• Proposes heuristics for optimal data collection in finite samples

Innovation

• Identifies drift and diffusion jointly
• Requires out-of-equilibrium observation
• Proposes heuristics for data collection