# Conditional Score-Based Modeling of Effective Langevin Dynamics

arXiv: https://arxiv.org/abs/2604.23952 · HTML: https://mlsanity.com/paper/2604.23952/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 7 of its 66 references that are in the map.
- First posted on arXiv 27 Apr 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 7 of 7)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Probing forced responses and causality in data-driven climate emulators: conceptual limitations and the role of reduced-order models](https://mlsanity.com/paper/2506.22552/) (2025)
- [Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification](https://mlsanity.com/paper/2412.00589/) (2024)
- [Response Theory via Generative Score Modeling](https://mlsanity.com/paper/2402.01029/) (2024)
- [Predicting Forced Responses of Probability Distributions via the Fluctuation-Dissipation Theorem and Generative Modeling](https://mlsanity.com/paper/2504.13333/) (2025)
- [Reduced-Order Modeling of Cyclo-Stationary Time Series Using Score-Based Generative Methods](https://mlsanity.com/paper/2508.19448/) (2025)

## Abstract (arXiv)

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 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. Across these tests, the inferred models preserve the invariant statistics while 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.