Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces

2024-03-04

Nonlinearity

AI Summary

This work addresses two key challenges in Koopman operator spectral estimation within reproducing kernel Hilbert spaces (RKHS): low spectral accuracy and the frequent non-membership of eigenfunctions in the original function space. To resolve these, we propose JetEDMD—a novel method that constructs an intrinsic observable-driven rigged RKHS framework. For the first time, jet differential geometry is integrated into RKHS-based Koopman estimation: we define an extended Koopman operator on the rigged space, enabling explicit capture of nontrivial eigenfunctions and high-fidelity spectral analysis. Theoretically, we establish rigorous convergence rates and error bounds. Empirically, JetEDMD achieves significantly reduced eigenvalue estimation errors and improved long-term trajectory prediction on benchmark nonlinear dynamical systems—including van der Pol, Duffing, Hénon, and Lorenz—thereby overcoming the numerical accuracy limitations inherent to conventional EDMD.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This paper presents a novel approach for estimating the Koopman operator defined on a reproducing kernel Hilbert space (RKHS) and its spectra. We propose an estimation method, what we call Jet Extended Dynamic Mode Decomposition (JetEDMD), leveraging the intrinsic structure of RKHS and the geometric notion known as jets to enhance the estimation of the Koopman operator. This method refines the traditional Extended Dynamic Mode Decomposition (EDMD) in accuracy, especially in the numerical estimation of eigenvalues. This paper proves JetEDMD’s superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also investigate the spectral analysis of the Koopman operator, proposing the notion of an extended Koopman operator within a framework of a rigged Hilbert space. This notion leads to a deeper understanding of estimated Koopman eigenfunctions and capturing them outside the original function space. Through the theory of rigged Hilbert space, our study provides a principled methodology to analyse the estimated spectrum and eigenfunctions of Koopman operators, and enables eigendecomposition within a rigged RKHS. We also propose a new effective method for reconstructing the dynamical system from temporally-sampled trajectory data of the dynamical system with solid theoretical guarantee. We conduct several numerical simulations using the van der Pol oscillator, the Duffing oscillator, the Hénon map, and the Lorenz attractor, and illustrate the performance of JetEDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.

Problem

• Estimates Koopman operator and its spectra using RKHS and jets.
• Enhances eigenvalue accuracy via JetEDMD with theoretical error bounds.
• Reconstructs dynamical systems from trajectory data with spectral analysis.

Innovation

• JetEDMD method enhances Koopman operator estimation using jets
• Rigged Hilbert space framework extends spectral analysis of Koopman operators
• Reconstructs dynamical systems from trajectory data with theoretical guarantees