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

TL;DR

This paper introduces a novel geometric approach to infer overdamped Langevin dynamics from sparse data by reformulating the problem as stochastic control, enabling accurate recovery of dynamics without high-frequency sampling.

Contribution

The authors develop a geometry-driven path augmentation framework that leverages invariant density geometry to infer stochastic dynamics from undersampled data, extending beyond conservative systems.

Findings

Accurately recovers stochastic dynamics from extremely undersampled data.

Outperforms existing methods in synthetic benchmarks.

Effectively incorporates geometric inductive biases into system identification.

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.