Conditional Score-Based Modeling of Effective Langevin Dynamics

A new data-driven calibration method for stochastic reduced-order models bypasses the need for short-time trajectory increments or repeated simulations, instead constraining drift and diffusion coefficients directly from finite-lag statistics, according to research posted to arXiv [1]. The framework, introduced by Ludovico T. Giorgini, rests on a relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density — the gradient of the logarithm of the transition density with respect to the initial state [1][2]. 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 [1][4]. This formulation allows the drift and diffusion structure to be constrained without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs [1][4]. Standard approaches to estimating effective Langevin dynamics often rely on short-time trajectory increments or state-space partitioning, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data [1][4]. The new method instead uses finite-lag statistical information, which can be extracted from stationary lagged pairs and is therefore suited to data that are temporally coarse, unevenly sampled, or noisy [4]. The central inverse relation is linear in the unknown mobility and provides the basis for a direct calibration procedure that does not require repeated forward integrations of candidate reduced models [4]. The work builds on score estimators in nonequilibrium response theory and stochastic modeling. In generalized fluctuation–dissipation theory, responses to weak perturbations are expressed through correlation functions evaluated in the unperturbed system, but for nonlinear nonequilibrium systems these formulas involve derivatives of the invariant density, which are difficult to estimate directly in high dimensions [3]. Learned stationary scores provide this information and have enabled data-driven response prediction for nonlinear and high-dimensional systems [3]. Prior score-based Langevin reduced models used a constant mobility matrix estimated from short-time coordinate autocorrelation, which could enforce only restricted short-time constraints [3]. The present work removes this restriction by introducing the conditional score as the object that links finite-lag transition statistics to a state-dependent mobility, turning score-based stochastic modeling into a framework where prescribed finite-lag dynamical constraints can be imposed directly from data [3]. The method was validated on three systems of increasing complexity: an analytically tractable Cox–Ingersoll–Ross diffusion, a two-dimensional nonequilibrium diffusion with affine multiplicative noise, and a periodic soft-spin stochastic Landau–Lifshitz chain [1]. Across these tests, the inferred models preserved the invariant statistics while accurately reproducing finite-lag dynamical correlations [1][2]. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties [1][2]. The paper was first submitted on 27 April 2026 and revised on 15 June 2026 [1].