Analyzing Koopman Operators in RKHS

Concetti Chiave

The author presents a novel method, Jet Dynamic Mode Decomposition (JetDMD), to enhance the estimation of the Koopman operator in reproducing kernel Hilbert spaces (RKHS) by leveraging intrinsic observables. This method refines traditional approaches and offers a solid theoretical foundation for performance improvement.

Sintesi

The paper introduces JetDMD as an innovative approach to estimate the Koopman operator in RKHS, providing superior accuracy in eigenvalue estimation. Through rigorous mathematical analysis, it establishes a principled methodology for spectral analysis and eigendecomposition within a rigged RKHS framework. The study showcases numerical simulations on various dynamical systems to illustrate JetDMD's effectiveness.
Key points:

Introduction of Jet Dynamic Mode Decomposition (JetDMD) for Koopman operator estimation.
Refinement of Extended Dynamic Mode Decomposition (EDMD) with intrinsic observables.
Theoretical foundation for improved accuracy in eigenvalue estimation.
Proposal of new methods for dynamical system reconstruction from trajectory data.
Application of JetDMD on chaotic systems like the Lorenz attractor.
Detailed mathematical analysis and proofs supporting the performance of JetDMD.

Statistiche

For any subset S ⊂ R, we denote by S>0 (resp. S≥0) the set of positive (resp. non-negative) elements of S.
Let λ1,...,λd be the eigenvalues of the Jacobian matrix d fp at p and let λ = (λ1,...,λd).
We define Vp,n = ∑ |α|≤n C·∂ α x k(x,·)|x=p.

Citazioni

"JetDMD provides much more precise depiction in numerical estimation of eigenvalues than EDMD."
"Jet is a geometric formulation providing a canonical object on a manifold."
"JetDMD achieves superior performance through simple truncation operations."

Approfondimenti chiave tratti da

Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces

by Isao Ishikaw... alle arxiv.org 03-06-2024

https://arxiv.org/pdf/2403.02524.pdf

Domande più approfondite

How does the concept of intrinsic observables impact traditional methods like EDMD

The concept of intrinsic observables has a significant impact on traditional methods like Extended Dynamic Mode Decomposition (EDMD). In EDMD, the estimation of the Koopman operator is based on a fixed set of observables within a specific function space. However, with intrinsic observables in rigged reproducing kernel Hilbert spaces (RKHS), we can refine this approach by leveraging the geometric notion known as jets. By constructing the space of intrinsic observables using jets at fixed points of dynamical systems, we can enhance the accuracy and performance of estimating the Koopman operator. This refinement provided by Jet Dynamic Mode Decomposition (JetDMD) allows for more precise numerical estimation of eigenvalues and captures essential components that may be missed by traditional methods like EDMD.

What are the practical implications of using JetDMD beyond numerical simulations

The practical implications of using JetDMD extend beyond numerical simulations to various applications in data-driven analysis and system identification. By incorporating intrinsic observables within a framework of RKHS and rigged Hilbert spaces, JetDMD offers a principled methodology to analyze estimated spectra and eigenfunctions of Koopman operators accurately. This method provides an alternative solution for issues such as spectral pollution, invariant subspaces, continuous spectrum, and chaotic dynamics commonly faced in dynamical systems analysis. Additionally, JetDMD enables effective reconstruction of complex dynamical systems from temporally-sampled trajectory data with theoretical guarantees for broad classes of analytic systems. The refined estimation provided by JetDMD enhances our ability to extract quantitative information from complex and chaotic dynamical systems across various fields.

How can the theory of rigged Hilbert space enhance our understanding of complex dynamical systems

The theory of rigged Hilbert space enhances our understanding of complex dynamical systems by providing a rigorous mathematical framework to analyze estimated eigenvectors and spectra in RKHS settings. Through the concept of extended Koopman operators defined within Gelfand triples constructed from intrinsic observables, we gain insights into eigendecompositions that go beyond traditional function spaces. The use of rigged Hilbert spaces allows us to capture essential characteristics outside the original function space where Koopman operators act while ensuring convergence results without imposing boundedness constraints on these operators. This theoretical foundation deepens our understanding and interpretation capabilities when analyzing complex dynamics through data-driven approaches like JetDMD.