# Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces arXiv: https://arxiv.org/abs/2403.02524 ยท HTML: https://mlsanity.com/paper/2403.02524/ ## Key facts - Cited by 9 papers in the map (data through 14 Oct 2026). - Builds on 22 of its 66 references that are in the map. - First posted on arXiv 4 Mar 2024. ## Builds on (most cited first, 22 of 22) - [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016) - [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021) - [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017) - [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017) - [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014) - [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017) - [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021) - [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022) - [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023) - [Active Learning of Dynamics for Data-Driven Control Using Koopman Operators](https://mlsanity.com/paper/1906.05194/) (2019) - [Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control](https://mlsanity.com/paper/1902.02827/) (2019) - [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023) - [Koopman-based lifting techniques for nonlinear systems identification](https://mlsanity.com/paper/1709.02003/) (2017) - [Learning Dynamics Models with Stable Invariant Sets](https://mlsanity.com/paper/2006.08935/) (2020) - [Estimating Koopman operators with sketching to provably learn large scale dynamical systems](https://mlsanity.com/paper/2306.04520/) (2023) - [Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables](https://mlsanity.com/paper/1808.10551/) (2018) - [Reproducing kernel Hilbert C*-module and kernel mean embeddings](https://mlsanity.com/paper/2101.11410/) (2021) - [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023) - [An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling](https://mlsanity.com/paper/1908.01047/) (2019) - [Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions](https://mlsanity.com/paper/1911.11992/) (2019) - [Koopman-based Control of a Soft Continuum Manipulator Under Variable Loading Conditions](https://mlsanity.com/paper/2002.01407/) (2020) - [Discriminant Dynamic Mode Decomposition for Labeled Spatio-Temporal Data Collections](https://mlsanity.com/paper/2102.09973/) (2021) ## Built on by (most cited first, 9 of 9) - [ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals](https://mlsanity.com/paper/2501.00701/) (2025) - [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025) - [Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator](https://mlsanity.com/paper/2412.16588/) (2024) - [Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation](https://mlsanity.com/paper/2603.01077/) (2026) - [Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics](https://mlsanity.com/paper/2603.06872/) (2026) - [Finite-dimensional approximations of push-forwards on locally analytic functionals](https://mlsanity.com/paper/2404.10769/) (2024) - [Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices](https://mlsanity.com/paper/2410.02199/) (2024) - [Spatially Aware Dictionary-Free Koopman Eigenfunction Identification for Modeling and Control](https://mlsanity.com/paper/2511.22648/) (2025) - [Generative Modeling through Koopman Spectral Analysis: An Operator-Theoretic Perspective](https://mlsanity.com/paper/2512.18837/) (2025) ## Abstract (arXiv) 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 analyze 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 H'enon map, and the Lorenz attractor, and illustrate the performance of JetEDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.