Score-based modeling of effective Langevin dynamics

We introduce a constructive framework to learn effective Langevin equations from stationary time series. Unlike conventional approaches that require iterative calibration to match target statistics, our construction guarantees the observed steady-state density by design and enforces short-lag coordinate-correlation constraints directly from data-so that the surrogate satisfies the targeted invariant measure and short-lag coordinate-correlation constraints from the outset, without trial-and-error tuning. The drift is parametrized in terms of the score function-the gradient of the logarithm of the steady-state distribution-and a constant mobility matrix whose symmetric part controls dissipation and diffusion and whose antisymmetric part encodes mean nonequilibrium circulation. The score is learned from samples using denoising score matching, while the mobility coefficients are inferred from short-lag correlation identities estimated via a clustering-based finite-volume discretization on a data-adaptive state-space partition. We validate the approach on low-dimensional stochastic benchmarks and on partially observed Kuramoto-Sivashinsky dynamics, where the resulting Markovian surrogate preserves the marginal invariant measure and captures the temporal correlations of the resolved variables. The resulting Langevin models define explicit reduced generators that enable efficient sampling and generation of synthetic trajectories with the imposed statistical and dynamical constraints, without direct simulation of the underlying full dynamics.