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

2026-06-12 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning, Engineering & Applied Sciences · Depth: Expert, extended

Summary

A new framework unifies temporal and geometric methods for inferring stochastic dynamics from sparse observations. This approach reformulates inference as a stochastic control problem, employing geometry-driven path augmentation guided by the system's invariant density to reconstruct likely trajectories. It accurately recovers overdamped Langevin system dynamics, even from extremely undersampled data, outperforming existing methods in synthetic benchmarks. The work demonstrates the effectiveness of incorporating geometric inductive biases into stochastic system identification, particularly for non-conservative systems such as the Van der Pol, Hopf, and Selkov glycolysis models, showing substantial improvement in estimating underlying stochastic dynamics in sparsely sampled, nonlinear contexts.

Key takeaway

For research scientists analyzing complex stochastic systems with sparse observational data, this method offers a robust solution. You should consider integrating geometry-aware path augmentation to overcome limitations of traditional temporal or purely geometric inference. This approach significantly improves drift estimation accuracy, especially for nonlinear, non-conservative systems, even when observations are extremely undersampled, providing more reliable insights into underlying dynamics.

Key insights

Reconciles temporal and geometric stochastic inference using geometry-driven path augmentation for sparse data.

Principles

• Incorporate geometric inductive biases into stochastic system identification methods.
• Augmented paths should lie near geodesic curves on the empirical manifold.

Method

Approximates Riemannian metric, computes geodesics between observations, then generates geometrically constrained diffusion bridges within an Expectation Maximisation framework.

In practice

• Accurately recover dynamics from extremely undersampled overdamped Langevin data.
• Infer stochastic dynamics in non-conservative systems (e.g., Van der Pol, Hopf).

Topics

• Langevin Dynamics
• Stochastic Differential Equations
• Geometric Inference
• Path Augmentation
• Sparse Data Analysis
• Riemannian Geometry
• Stochastic Control

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.