Finding Koopman Invariant Subspaces via Personalized PageRank

2026-05-23

AI Summary

This work addresses the key challenge in data-driven Koopman operator approximation: automatically identifying a finite subdictionary from a high-dimensional observation dictionary that spans a Koopman-invariant subspace. It introduces personalized PageRank to this problem for the first time, leveraging the zero-block structure of the extended dynamic mode decomposition (EDMD) matrix to precisely select observable subsets corresponding to invariant subspaces on the row-normalized EDMD matrix. Theoretically, it establishes a rigorous connection between zero-block structure and Koopman invariance and provides end-to-end detection guarantees under finite-sample settings via matrix perturbation theory and concentration inequalities. Experiments demonstrate that the method consistently identifies compact, interpretable dictionaries and achieves high-accuracy dynamical predictions across benchmark systems, including Duffing, Van der Pol, Lorenz, and the three-well Ramachandran potential model.

Abstract

Selecting a finite dictionary of observables whose span is Koopman-invariant is a central challenge in data-driven Koopman operator approximation. We address this problem by exploiting zero-block structure in Extended Dynamic Mode Decomposition (EDMD) matrices. We show that any sub-dictionary whose span is Koopman-invariant induces an exact zero block in the EDMD matrix, even for finite data. We then show that such blocks can be detected by applying PageRank to a row-normalized EDMD matrix constructed from a large initial dictionary. The theory extends to approximately invariant subspaces and yields stronger guarantees for personalized PageRank (PPR) when the seed observables lie inside the target block and reach all observables in that block. Combining EDMD concentration bounds with PageRank perturbation theory gives end-to-end detection guarantees with $O(1/\sqrt{M})$ finite-sample scaling and explicit constants. More generally, without assuming an invariant subspace exists, high PPR mass on a sub-dictionary controls discounted multi-step leakage from the seed observables. Numerical experiments on the Duffing oscillator, Van der Pol oscillator, Lorenz system, and a three-well Ramachandran potential suggest that the method identifies compact, interpretable dictionaries with accurate predictions.

Problem

• Koopman operator
• invariant subspace
• dictionary selection
• Extended Dynamic Mode Decomposition
• observables

Innovation

• Koopman operator
• Extended Dynamic Mode Decomposition
• Personalized PageRank
• invariant subspace
• finite-sample guarantee