From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints

TL;DR

This work tackles learning overdamped Langevin dynamics from sparsely sampled trajectories by unifying temporal and geometric viewpoints through stochastic control and geometry-guided path augmentation. It introduces a nonparametric, data-driven Riemannian metric derived from the invariant density to compute geodesics that constrain latent diffusion bridges, enabling accurate drift inference via an EM framework. Across model systems, the approach outperforms existing methods, especially at large inter-observation intervals and in non-conservative settings, and demonstrates robustness to moderate noise-misspecification. By embedding geometric inductive biases into stochastic system identification, the method offers a scalable, principled way to recover dynamics from undersampled data with practical implications for physics, biology, and beyond.

Abstract

How can we learn the laws underlying the dynamics of stochastic systems when their trajectories are sampled sparsely in time? Existing methods either require temporally resolved high-frequency observations, or rely on geometric arguments that apply only to conservative systems, limiting the range of dynamics they can recover. Here, we present a new framework that reconciles these two perspectives by reformulating inference as a stochastic control problem. Our method uses geometry-driven path augmentation, guided by the geometry in the system's invariant density to reconstruct likely trajectories and infer the underlying dynamics without assuming specific parametric models. Applied to overdamped Langevin systems, our approach accurately recovers stochastic dynamics even from extremely undersampled data, outperforming existing methods in synthetic benchmarks. This work demonstrates the effectiveness of incorporating geometric inductive biases into stochastic system identification methods.

Figures (5)

• Figure 1: Temporal and geometric perspectives for discovering stochastic dynamics and proposed inference with geometrically guided augmentation. (A.) Temporal methods consider the time-ordering of observations $\{\boldsymbol{\mathcal{O}}_k\}^K_{k=1}$ (purple dots) to approximate the drift $\mathbf{f}(\mathbf{x})$ with conditional rescaled state increments $\hat{\mathbf{f}}(\mathbf{x}) = \langle \frac{\text{d} \mathbf{X}_t}{\tau} | \mathbf{X}_t=\mathbf{x} \rangle$. (B.) Geometric methods assume a conservative drift $\mathbf{f}(\mathbf{x})=-\nabla V(\mathbf{x})$ as the gradient of a potential. (C.) With increasing inter-observation interval $\tau$ performance of temporal methods degrades because Euclidean distances ignore the curvature of the latent continuous path between consecutive observations. (D.) Path augmentation alternates between state estimation - by sampling diffusion bridges for each inter-observation interval - and drift inference. (E.) Commonly used path augmentation methods employ Brownian or Ornstein-Uhlenbeck bridges that increasingly deviate from the unobserved path as $\tau$ grows. (lower) Illustration of the ground truth (neon green) and geodesic (magenta) continuous path between two observations and of that assumed during inference with Gaussian likelihood (yellow line). (F.) Geometrically guided augmentation approximates first the metric induced by the invariant density, constructs geodesics connecting consecutive observations, and samples geometrically constrained diffusion bridges.
• Figure 2: Geometry-aware path augmentation improves drift inference after two iterations. Estimated (red) vs. true (grey) force field with a.) Gaussian likelihood, b.) after one, and c.) after two augmentations. (Insets) True vs. estimated angles at grid points. d.) Weighted (by observation density) root mean square error (wRMSE) vs. inter-observation interval $\tau$ for different noise levels $\sigma=\{0.25,0.5\}$ for drift estimated with a Gaussian likelihood (gaus-circles), after first augmentation (1st-triangles), and after second augmentation (2nd-squares) for $T=500$ (time units). e.) wRMSE across iterations for the presented example. f.) wRMSE vs. noise amplitude $\sigma$ for different trajectory durations ${T=\{500,1000\}}$ (time units) for inter-observation interval $\tau=240$ ($dt$). Markers in d.) and f.) indicate augmentation steps. Error bars: one standard deviation over five independent runs.
• Figure 3: Comparison of geometry-aware inference against inference with Ornstein-Uhlenbeck augmentation. Weighted root mean square error (wRMSE) vs. different inter-observation intervals $\tau$ for different noise amplitudes for moderate inter-observation intervals with a.)$\sigma=0.25$ and b.)$\sigma=0.50$, and for large inter-observation intervals with c.)$\sigma=0.50$ and d.)$\sigma=0.75$, where only one observation per oscillation period is available. Error bars indicate one standard deviation over five independent runs.
• Figure 4: Geometry-aware inference provides accurate drift estimation for different empirical manifold geometries resulting from different parameter regimes of the Van der Pol system.(a.-b.)Empirical manifold for the Van der Pol system with different $\mu$ parameters. Notice the different scales on the axes. (c.-d.) Inference performance of the proposed framework against inter-observation interval $\tau$. Error bars indicate one standard deviation over five independent runs.
• Figure 5: Small noise misestimation has small impact on estimation accuracy. Weighted root mean square error (wRMSE) vs. noise amplitude $\sigma$ employed in the augmentation for different inter-observation intervals with a.)$\tau=160\,dt$b.)$\tau=200\,dt$, c.)$\tau=240\,dt$d.)$\tau=280\,dt$. Pink-purple lines correspond to estimation with total simulation length $T=500$ time units, and blue markers correspond to total simulation length of $T=1000$ time units. Red dotted line identifies the noise amplitude employed in the simulation of the observations.