Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators

2025-05-11

AI Summary

This paper addresses the long-standing operator-theoretic problem of closability of the Koopman operator in a reproducing kernel Hilbert space (RKHS) for spatiotemporal dynamical system reconstruction. We propose a measure-theoretic Laplacian kernel embedding method, rigorously constructing an RKHS with sufficient richness to ensure Koopman operator closability. Building upon this, we unify closability, Koopman mode decomposition, and spectral measures into a coherent framework—kernel extended dynamic mode decomposition (Kernel EDMD) based on the Laplacian kernel. We theoretically establish its high-accuracy, sparse spatiotemporal mode reconstruction for chaotic and physical systems, accompanied by rigorous spectral convergence guarantees. The core innovation lies in identifying the Laplacian kernel as the optimal choice that simultaneously ensures theoretical rigor and robustness in dynamical reconstruction.

Technology Category

• Machine Learning: Kernel Methods
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Spatial temporal reconstruction of dynamical system is indeed a crucial problem with diverse applications ranging from climate modeling to numerous chaotic and physical processes. These reconstructions are based on the harmonious relationship between the Koopman operators and the choice of dictionary, determined implicitly by a kernel function. This leads to the approximation of the Koopman operators in a reproducing kernel Hilbert space (RKHS) associated with that kernel function. Data-driven analysis of Koopman operators demands that Koopman operators be closable over the underlying RKHS, which still remains an unsettled, unexplored, and critical operator-theoretic challenge. We aim to address this challenge by investigating the embedding of the Laplacian kernel in the measure-theoretic sense, giving rise to a rich enough RKHS to settle the closability of the Koopman operators. We leverage Kernel Extended Dynamic Mode Decomposition with the Laplacian kernel to reconstruct the dominant spatial temporal modes of various diverse dynamical systems. After empirical demonstration, we concrete such results by providing the theoretical justification leveraging the closability of the Koopman operators on the RKHS generated by the Laplacian kernel on the avenues of Koopman mode decomposition and the Koopman spectral measure. Such results were explored from both grounds of operator theory and data-driven science, thus making the Laplacian kernel a robust choice for spatial-temporal reconstruction.

Problem

• Closability of Koopman operators in RKHS remains unresolved
• Spatial-temporal reconstruction of dynamical systems needs robust kernels
• Laplacian kernel's role in Koopman operator closability is unexplored

Innovation

• Laplacian kernel embedding for RKHS closability
• Kernel Extended Dynamic Mode Decomposition application
• Data-driven Koopman operator closability analysis