From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints
This work addresses a challenge in stochastic system identification for researchers, offering a method that works with sparse data, though it is incremental as it builds on geometric approaches.
The paper tackles the problem of learning stochastic dynamics from sparse time-sampled trajectories, presenting a framework that uses geometry-driven path augmentation to accurately recover overdamped Langevin dynamics, outperforming existing methods in synthetic benchmarks.
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.