Geometric Dictionary Learning of Dynamical Systems with Optimal Transport

2026-05-18

AI Summary

This work addresses the limitation of conventional dynamical system operator estimation methods, which treat each system in isolation and fail to exploit shared dynamical structures. The authors propose DOODL, a novel framework that integrates dictionary learning with optimal transport for the first time, under the assumption that related systems approximately lie on a low-dimensional manifold in the spectral operator space. By learning a shared basis of spectral dynamical atoms, DOODL geometrically models this manifold to yield compact, interpretable system embeddings and effectively regularizes operator estimation from short or partial observations. Evaluated on Langevin dynamics and turbulent plasma simulations, DOODL reduces estimation errors by one to two orders of magnitude compared to traditional approaches under low-data regimes, substantially improving the recovery accuracy of dominant spectral dynamical structures.

Abstract

Learning dynamical systems through operator-theoretic representations provides a powerful framework for analyzing complex dynamics, as spectral quantities such as eigenvalues and invariant structures encode characteristic time scales and long-term behavior. However, dynamical operators are typically estimated independently for each system, preventing the discovery of shared structure across related dynamics. To address this limitation, we posit that related dynamical systems lie near a low-dimensional manifold in spectral operator space. Based on this hypothesis, we introduce DOODL (Dynamical OperatOr Dictionary Learning), a framework that learns a dictionary of characteristic spectral dynamics whose combinations approximate this manifold and yield compact, interpretable embeddings of individual systems. Beyond representation learning, DOODL enables fast and interpretable operator estimation from short and partially observed trajectories by constraining the estimation to the learned operator manifold. Experiments on metastable Langevin dynamics and turbulent plasma simulations demonstrate that DOODL scales to highly complex multiscale regimes while capturing characteristic spectral structure governing the dynamics rather than merely fitting trajectories, achieving errors one to two orders of magnitude lower than independent operator estimation methods in challenging low-data regimes.

Problem

• dynamical systems
• operator-theoretic representations
• spectral structure
• shared structure
• dictionary learning

Innovation

• Dictionary Learning
• Operator-theoretic Representation
• Optimal Transport
• Spectral Dynamics
• Low-dimensional Manifold