Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators

TL;DR

This paper introduces a kernel-based approach using Laplacian kernels to address the challenge of reconstructing dynamical systems through Koopman operators in RKHS, providing both theoretical and empirical validation.

Contribution

It proposes a novel method leveraging Laplacian kernels for Koopman operator approximation, ensuring closability in RKHS, and demonstrates its effectiveness in spatial-temporal reconstruction.

Findings

Successful reconstruction of dynamical systems using the proposed kernel method

Theoretical proof of Koopman operator closability in the RKHS generated by the Laplacian kernel

Empirical validation across diverse dynamical systems

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.