[Paper Review] Koopman Operator Family Spectrum for Nonautonomous Systems - Part 1

Senka Mačešić, Nelida Črnjarić-Žic+1 more|arXiv (Cornell University)|Mar 21, 2017

Model Reduction and Neural Networks14 references3 citations

TL;DR

This paper proposes a data-driven algorithm for computing the spectrum of the Koopman operator family in linear non-autonomous dynamical systems by leveraging the fundamental matrix and adaptive observable selection. It reveals that standard Arnoldi-like methods fail due to time-derivative-induced errors in eigenvalue approximations, and introduces Algorithm 2 using nonlinear conjugate observables to achieve high accuracy, validated across hybrid and continuous-time systems with exact eigenvalue recovery.

ABSTRACT

For every non-autonomous system, there is the related family of Koopman operators $\mathcal{K}^{(t,t_0)}$, parameterized by the time pair $(t,t_0)$. In this paper we are investigating the time dependency of the spectral properties of the Koopman operator family in the linear non-autonomous case and we propose an algorithm for computation of its spectrum from observed data only. To build this algorithm we use the concept of the fundamental matrix of linear non-autonomous systems and some specific aspects of Arnoldi-like methods. In particular, we use Arnoldi-like methods on local data stencils, we exploit the information contained in the Krylov subspace projection error, and discover limitations in the application of Arnoldi-like methods to cases with continous time dependency. We present results of this data-driven algorithm on various linear non-autonomous systems, hybrid as well as continuous in time. In all the examples comparison with exact eigenvalues and eigenfunctions shows excellent performance of the proposed algorithm.

Motivation & Objective

• To develop a data-driven algorithm for computing the Koopman operator family spectrum in linear non-autonomous systems.
• To identify and resolve the failure of standard Arnoldi-like methods in approximating time-dependent Koopman operators due to time-derivative-induced eigenvalue errors.
• To formalize a method that detects switching events in hybrid systems using Krylov subspace projection error.
• To design a new observable transformation strategy that enables accurate spectral computation in continuously time-dependent systems.
• To validate the proposed algorithms on both hybrid and continuous-time linear non-autonomous systems with exact eigenvalue comparisons.

Proposed method

• The Koopman operator family is linked to the fundamental matrix family of the underlying linear non-autonomous system via a theoretical connection established in Theorem 1.
• Algorithm 1 uses Krylov subspace projection error on moving stencils to detect time points of matrix switching in hybrid systems.
• Arnoldi-like methods are applied on local data stencils, with error analysis revealing that projection error signals switching events in the governing matrix.
• Algorithm 2 introduces nonlinear conjugate observables of the form (54), such as $ u_1 = rac{x_1 + i x_3}{ ext{norm}} $, to isolate real or imaginary parts of eigenvalues.
• The new observables ensure that two-snapshot stencils contain sufficient information for accurate Koopman eigenvalue computation, bypassing the limitations of constant-matrix assumptions.
• The method exploits the fact that Arnoldi-like methods fail when the underlying matrix is time-dependent, especially when eigenvalue derivatives are non-zero.

Experimental results

Research questions

• RQ1 Why do standard Arnoldi-like methods fail to accurately approximate the Koopman spectrum in time-dependent non-autonomous systems?
• RQ2 How can the Krylov subspace projection error be used to detect switching events in hybrid linear non-autonomous systems?
• RQ3 What observable transformations are necessary to enable accurate Koopman spectrum computation in continuously time-dependent systems?
• RQ4 Can a data-driven algorithm recover the exact Koopman eigenvalues and eigenfunctions for linear non-autonomous systems from sparse time-series data?
• RQ5 What is the nature of the error in Koopman eigenvalue approximations when the underlying matrix is time-dependent, and how does it scale with system complexity?

Key findings

• The proposed Algorithm 2, using nonlinear conjugate observables, achieves near-perfect recovery of exact Koopman eigenvalues in both hybrid and continuous-time systems, with no visible offset from exact values in numerical results.
• Standard DMD and Arnoldi-like methods produce significant errors in eigenvalue approximation for time-dependent systems, particularly in real parts when eigenvalues are purely imaginary, due to non-zero time derivatives.
• The error in eigenvalue approximation is proportional to the time derivative of the eigenvalues, and does not diminish with reduced time step, indicating a fundamental incompatibility with time-varying matrices.
• Krylov subspace projection error increases at switching points in hybrid systems, enabling detection of matrix transitions and forming the basis of Algorithm 1.
• The use of observables like $ u_1 = rac{x_1 + i x_3}{ ext{norm}} $ successfully decouples real and imaginary components, allowing accurate reconstruction of eigenvalues from minimal data stencils.
• All numerical examples, including coupled oscillators with continuous frequency change, confirm that the proposed algorithms outperform standard methods, with results matching exact eigenvalues within numerical precision.