Conditional Score-Based Modeling of Effective Langevin Dynamics

TL;DR

This paper introduces a novel, data-driven method for calibrating stochastic reduced-order models using conditional scores, enabling efficient and accurate modeling of complex systems' dynamics from finite-lag statistics.

Contribution

It develops a new relationship between model coefficients and the conditional score, allowing direct constraint of drift and diffusion from data without trajectory differentiation or state-space partitioning.

Findings

The method accurately infers models that preserve invariant statistics.

It reproduces finite-lag dynamical correlations effectively.

The approach is scalable for learning stochastic models from data.

Abstract

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on analytically tractable and data-driven nonequilibrium diffusions, demonstrating that the inferred models preserve the invariant statistics while accurately reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.