Score-Based Modeling of Effective Langevin Dynamics

TL;DR

The paper tackles the problem of constructing reduced Markovian Langevin models from stationary time series that faithfully reproduce the target invariant measure $p_{\mathrm{ss}}$ and temporal correlations $\mathbf{C}(\tau)$. It introduces a score-based drift representation $\vec{F}(\vec{x}) = \bm{\Sigma}\bm{\Sigma}^T\vec{\nabla}\ln p_{\mathrm{ss}} + \nabla\cdot\mathbf{R} + \mathbf{R}\vec{\nabla}\ln p_{\mathrm{ss}}$, with $\mathbf{R}$ antisymmetric, and learns the steady-state score $\vec{s}(\vec{x})=\nabla\ln p_{\mathrm{ss}}(\vec{x})$ via denoising score matching while inferring diffusion from short-lag correlation identities through a data-adaptive rate-matrix discretization. This construction guarantees the invariant measure a priori and enforces dynamical constraints directly from data, avoiding iterative calibration loops. The method is validated on low-dimensional benchmarks and a partially observed Kuramoto–Sivashinsky system, yielding reduced generators that enable efficient sampling of synthetic trajectories with correct marginal and temporal statistics. The resulting framework provides interpretable decomposition into symmetric (diffusive) and antisymmetric (circulatory) components, and offers a scalable path to stochastic closures for high-dimensional, partially observed systems. Key innovations include the score-based parameterization of the drift, the Stein-identity–driven link between short-time dynamics and the drift/diffusion pair, and a practical, data-driven estimator for the coarse generator on a data-adaptive partition. The approach delivers explicit reduced SDEs that reproduce non-Gaussian steady states and nontrivial time correlations, with potential impact on turbulence modeling, climate closures, and other complex systems where partial observability and multiscale effects prevail.

Abstract

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 temporal correlations directly from data -- so that the surrogate satisfies the targeted invariant measure and correlation constraints from the outset, without trial-and-error tuning. The drift is parameterized 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 captures the marginal invariant measure and temporal correlations of the resolved modes. The resulting Langevin models define explicit reduced generators that enable efficient sampling and generation of synthetic trajectories with correct temporal statistics, without direct simulation of the underlying full dynamics.

Figures (4)

• Figure 1: Kuramoto--Sivashinsky reduced model validation.Left: Spatiotemporal evolution (Hovmöller plots) of the spectral modes for the reduced Langevin model (top) and the ground truth data (bottom). Center: Marginal invariant distribution (PDF) and autocorrelation function (ACF) for a representative mode (all modes are statistically equivalent due to periodic boundary conditions), comparing the model (red) with data (blue). Right: Joint probability densities $p(x_i, x_{i+k})$ for lags $k=1$, $k=2$, and $k=3$ (left to right), showing the ability of the model to capture spatial correlations between modes.
• Figure 2: Learned operators for the KS equation. Comparison of the learned linear operators. From left to right: the full drift matrix $\bm{\Phi}$, the symmetric part $\bm{\Phi}_S$ (determining the diffusion tensor), the antisymmetric part $\bm{\Phi}_A$, the noise amplitude matrix $\bm{\Sigma}$ (lower triangular), and the score-position correlation matrix $\langle \vec{s}\,\vec{x}^T \rangle$ verifying Stein's identity ($\approx -\bm{I}$).
• Figure 3: One-dimensional nonlinear SDE.First row: Time-series comparison between the original system (Observed) and the reconstructed dynamics (Model) using the DSM-estimated score function and $\bm{\Phi}$. The same noise realization was used to generate both time series. Second row: Comparison of the observed marginal PDFs (blue) with the reconstructed PDFs (red) obtained from the Langevin equation using the KGMM-estimated score function and $\bm{\Phi}$. Third row: Comparison of the observed ACFs (blue) with the reconstructed ACFs (red).
• Figure 4: Two-dimensional asymmetric potential system.First row: Comparison of trajectories between the original system (Observed) and the reconstructed dynamics (Model) using the KGMM-estimated score function and $\bm{\Phi}$. The same noise realization was used to generate both time series. Second row: Comparison of the observed marginal PDFs (blue) with the reconstructed PDFs (red) for each variable. Third row: Comparison of the observed ACFs (blue) with the reconstructed ACFs (red) for each variable.