Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators

TL;DR

The paper tackles spatial-temporal reconstruction of dynamical systems under irregular, sparse data by embedding Koopman operator analysis in a reproducing kernel Hilbert space (RKHS) defined by kernels. It introduces Lap-KeDMD, which uses the Laplacian kernel embedded as an $L^2$-measure to form an RKHS $H_{ ext{L}}$ and proves that Koopman operators on $H_{ ext{L}}$ are closable, unlike for the GRBF kernel where closability can fail. The authors provide empirical evidence across seven datasets showing that Lap-KeDMD yields faithful spatio-temporal reconstructions and supply a theoretical operator-theoretic foundation via spectral measures and faithful Koopman-mode decomposition within $H_{ ext{L}}$. The work demonstrates that the Laplacian kernel not only supports robust data-driven reconstruction under irregular sampling but also offers rigorous operator-analytic justification for its effectiveness, suggesting a principled kernel choice for KeDMD in challenging data regimes.

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.

Figures (27)

• Figure 1: The paradigm of interpret-ability of a dynamical system through Kernel Extended Dynamic Mode Decompositions. We will incorporate the Laplacian kernel to overcome our \ref{['aim']} whose Koopman operator theoretic inference is given in the last couple of equations capturing Koopman mode decomposition difference.
• Figure 1: The lifting perspective of the Koopman operators.
• Figure 1: An illustration of notion of observables for the Burger Equation's data (middle) when we chose the dictionary of Laplacian kernel (left) and GRBF kernel (right).
• Figure 1: Schematic presentation of important aspects of GRBF kernel in relation with SE covariance function and the Gaussian measure. GRBF kernel evaluated along $(\bm{z},-\bm{z})$ provides the GRBF measure which leads to RKHS Segal-Bargmann-Fock Space with exponential dot product kernel as the reproducing kernel.
• Figure 1: An illustration of the spectral measure approximations of the data-driven Koopman operators over $L^2(\Omega)$ due to the closability of the function theoretic Koopman operators over the RKHS $H_{\mathds{L}}$. Note that, this is under an assumption that the operator-theoretic Koopman operator has symbol linear in $\mathbb{C}^D$.

Theorems & Definitions (27)

• Definition 2.2: Koopman Operators
• Definition 2.3
• Definition 2.4
• Definition 2.5: Kernel Function
• Definition 2.6
• Example 2.7
• Definition 2.8: Reproducing Kernel Hilbert Space
• Theorem 2.9: Moore-Aronszajn Theorem aronszajn1950theory
• Theorem 4.1: Bochner's Theorem
• Theorem 4.2