Conditional Score-Based Modeling of Effective Langevin Dynamics

A new paper from Ludovico T. Giorgini, published on arXiv (2604.23952), introduces a data-driven calibration method for stochastic reduced-order models, specifically targeting effective Langevin dynamics. The key innovation is a novel identity that connects the drift and diffusion coefficients of a stochastic reduced model to the conditional score of the finite-time transition density—defined as the gradient of the log transition density with respect to the initial state. This allows the model coefficients to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration. The result is a least-squares fitting problem over stationary lagged pairs, which is computationally efficient and scalable.

The method is validated on both analytically tractable and data-driven nonequilibrium diffusions, demonstrating that the inferred models preserve invariant statistics while accurately reproducing finite-lag dynamical correlations. This approach addresses common challenges in stochastic modeling, such as high-dimensional systems, coarse temporal sampling, or unevenly sampled data. By bypassing expensive simulation steps, it provides a scalable route for learning stochastic reduced-order models that reproduce prescribed statistical and dynamical properties. The work sits at the intersection of machine learning (stat.ML, cs.LG) and chaotic dynamics (nlin.CD), and has potential applications in physics, finance, and engineering where complex systems are often approximated by stochastic differential equations.