[Paper Review] On the Spectral Equivalence of Koopman Operators through Delay Embedding

Yoshihiko Susuki, Kyoichi Sako+1 more|arXiv (Cornell University)|Jun 3, 2017

Model Reduction and Neural Networks11 references3 citations

TL;DR

This paper establishes spectral equivalence between the Koopman operator of a measure-preserving dynamical system and its delay-embedded reconstruction, proving that eigenvalues and eigenfunctions of the original system can be reliably estimated via Extended Dynamic Mode Decomposition (EDMD) applied to time-series data. The key result is that the Koopman operator on the reconstructed space is unitarily equivalent to the original, ensuring spectral fidelity under delay embedding for compact, measure-preserving systems.

ABSTRACT

We provide one theorem of spectral equivalence of Koopman operators of an original dynamical system and its reconstructed one through the delay-embedding technique. The theorem is proved for measure-preserving maps (e.g. dynamics on compact attractors) and provides a mathematical foundation of computing spectral properties of the Koopman operators by a combination of extended dynamic mode decomposition and delay-embedding.

Motivation & Objective

• To establish a mathematical foundation for estimating Koopman operator spectra from scalar time-series using delay-embedding techniques.
• To resolve the fundamental question of whether spectral properties of the Koopman operator are preserved under delay-embedding for measure-preserving dynamical systems.
• To validate the use of Extended Dynamic Mode Decomposition (EDMD) on reconstructed state dynamics as a reliable method for approximating the Koopman spectrum of the original system.
• To formalize the relationship between the Koopman operators of the original and reconstructed dynamical systems through unitary equivalence.

Proposed method

• Utilizes Takens' embedding theorem to construct a delay-embedded reconstruction of the state space using a single observable and its time-lagged versions.
• Defines a homeomorphism φ from the original manifold M to the reconstructed space Φ(M) ⊂ ℝ^(2m+1), enabling a commutative diagram between original and reconstructed maps.
• Introduces a composition operator C_φ: L₂(φ(M)) → L₂(M) defined by g ↦ g∘φ, which maps observables on the reconstructed space back to the original space.
• Proves that C_φ is unitary under the measure-preserving condition, ensuring isometric preservation of inner products and spectral structure.
• Establishes the commutative relation C_φ ∘ U_~T = U_T ∘ C_φ between the Koopman operators on the original and reconstructed spaces.
• Demonstrates that spectral equivalence follows from the unitary equivalence of the Koopman operators via the composition operator C_φ.

Experimental results

Research questions

• RQ1 Does the Koopman operator spectrum of a measure-preserving dynamical system remain equivalent under delay-embedding reconstruction?
• RQ2 Can Extended Dynamic Mode Decomposition (EDMD) applied to delay-embedded data reliably estimate the Koopman spectrum of the original system?
• RQ3 What is the mathematical relationship between the Koopman operators of the original and reconstructed dynamical systems via delay embedding?
• RQ4 Under what conditions is the composition operator C_φ between reconstructed and original spaces unitary, ensuring spectral equivalence?

Key findings

• The Koopman operator U_T on the original system and U_~T on the reconstructed system are spectrally equivalent when T is measure-preserving.
• The composition operator C_φ: g ↦ g∘φ is unitary, ensuring that the inner product structure and spectral properties are preserved between the original and reconstructed spaces.
• The spectral equivalence is guaranteed by the unitary equivalence of the Koopman operators via the homeomorphism φ and the measure-preserving property of T.
• The result validates the use of EDMD on delay-embedded data to estimate the eigenvalues and eigenfunctions of the original Koopman operator.
• Eigenfunctions and eigendistributions of the original Koopman operator can be reconstructed via the inverse of C_φ, though φ and C_φ are typically unknown in practice.