从稀疏数据中学习随机系统动态，利用几何先验提升恢复精度
From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints
- 将推断重构为随机控制问题，通过系统不变密度引导路径补全
- 在极低采样率下仍能准确恢复过阻尼Langevin动力学
- 适合缺乏高频观测的物理系统建模，尤其适用于非保守系统
当随机系统的轨迹仅稀疏采样时，如何学习其内在动力学规律？现有方法或需高频率时间序列数据，或仅适用于保守系统。本文提出新框架，将推断重构为随机控制问题，利用系统不变密度引导几何驱动的路径补全，无需预设参数模型即可重建可能轨迹并推断底层动态。应用于过阻尼Langevin系统时，即使在极端欠采样条件下仍能精确恢复随机动力学，在合成基准测试中优于现有方法。该研究证明了将几何归纳偏置融入随机系统识别的有效性。
原文摘要 · Abstract (English)
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.
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