On the algebra of Koopman eigenfunctions and on some of their infinities

2026-04-23

AI Summary

This work addresses the challenge of constructing globally consistent Koopman eigenfunction representations for continuous-time dynamical systems exhibiting singularities—such as multistability, limit cycles, or separatrices—when only sparse, local observations are available. Conventional approaches struggle to achieve this efficiently. Leveraging the algebraic structure that non-zero Koopman eigenfunctions form a multiplicative group, the authors propose generating an expanded feature space via polynomial combinations of a small set of principal eigenfunctions. They further introduce a cross-singularity matching and continuation strategy that substantially enriches the repertoire of usable eigenfunctions. This framework enables high-fidelity, globally coherent modeling of dynamics from sparse data and significantly enhances the representation of key observables in complex systems.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

For continuous-time dynamical systems with reversible trajectories, the nowhere-vanishing eigenfunctions of the Koopman operator of the system form a multiplicative group. Here, we exploit this property to accelerate the systematic numerical computation of the eigenspaces of the operator. Given a small set of (so-called ``principal'') eigenfunctions that are approximated conventionally, we can obtain a much larger set by constructing polynomials of the principal eigenfunctions. This enriches the set, and thus allows us to more accurately represent application-specific observables. Often, eigenfunctions exhibit localized singularities (e.g. in simple, one-dimensional problems with multiple steady states) or extended ones (e.g. in simple, two-dimensional problems possessing a limit cycle, or a separatrix); we discuss eigenfunction matching/continuation across such singularities. By handling eigenfunction singularities and enabling their continuation, our approach supports learning consistent global representations from locally sampled data. This is particularly relevant for multistable systems and applications with sparse or fragmented measurements.

Problem

• Koopman eigenfunctions
• singularities
• multistable systems
• global representation
• dynamical systems

Innovation

• Koopman operator
• eigenfunction algebra
• singularity continuation
• multistable systems
• global representation