Geometric Dictionary Learning of Dynamical Systems with Optimal Transport

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

DOODL (Dynamical OperatOr Dictionary Learning) is a novel framework designed to learn and represent dynamical systems by identifying shared spectral structures across related dynamics. It hypothesizes that related systems reside on a low-dimensional manifold within a spectral operator space. DOODL leverages optimal transport and Riemannian dictionary learning to construct compact, interpretable operator embeddings. This approach enables efficient and accurate operator estimation from short, partially observed trajectories, outperforming independent estimation methods by one to two orders of magnitude in low-data regimes. Experiments on metastable Langevin dynamics and turbulent plasma simulations demonstrate DOODL's ability to capture characteristic spectral structures, facilitating tasks like early system identification and detection of regime shifts in complex, multiscale environments.

Key takeaway

For AI Scientists and Research Scientists working with complex dynamical systems from limited data, DOODL offers a powerful approach to overcome data scarcity. By learning a low-dimensional spectral manifold, you can achieve significantly more accurate operator estimations and enable rapid system identification or regime shift detection, even with short or partial trajectories. Consider integrating DOODL to enhance the interpretability and data efficiency of your dynamic modeling efforts.

Key insights

DOODL learns shared spectral manifolds of dynamical systems for robust, data-efficient operator estimation and system identification.

Principles

• Dynamical systems can be represented as low-dimensional manifolds in spectral operator space.
• Optimal transport provides a robust geometry for comparing and interpolating dynamical operators.
• Constraining operator estimation to a learned manifold improves accuracy in low-data regimes.

Method

DOODL minimizes an optimal transport loss to learn a dictionary of characteristic spectral dynamics, using a projection-based reconstruction model and Riemannian gradient optimization for dictionary and coefficient updates.

In practice

• Use DOODL for early system identification from short trajectories.
• Apply DOODL to detect regime shifts in complex dynamics.
• Employ DOODL for reduced-order representation of turbulent systems.

Topics

• Dynamical Systems
• Operator-Theoretic Representations
• Dictionary Learning
• Optimal Transport
• Spectral Decomposition

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.