Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators

TL;DR

Rigged DMD provides a data-driven framework to compute generalized eigenfunctions of Koopman operators in the presence of continuous spectra. By first obtaining a unitary, data-driven approximation of the Koopman operator via mpEDMD and then sampling its resolvent with high-order kernels, the method constructs wave-packet approximations to generalized eigenfunctions, enabling coherent modal decompositions beyond traditional DMD. The authors establish convergence results for the resolvent, generalized eigenfunctions, and spectral measures, and introduce a natural rigged Hilbert space construction via time-delay embedding to accommodate a broad class of systems. The approach is demonstrated on systems with Lebesgue spectra, integrable Hamiltonian dynamics, the Lorenz system, and high-Reynolds-number fluid flows, illustrating robustness, convergence, and applicability to complex, continuous-spectrum dynamics. This framework opens avenues for modal analysis, reduced-order modeling, and control in nonlinear systems where continuous spectral components play a central role.

Abstract

We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering the evolution of observables, Koopman operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis. While powerful, traditional Dynamic Mode Decomposition (DMD) techniques often struggle with continuous spectra. Rigged DMD addresses these challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions using snapshot data from the system's evolution. At its core, Rigged DMD builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition with high-order kernels for smoothing. This provides a robust decomposition encompassing both discrete and continuous spectral elements. We derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures. Additionally, we propose a novel framework for constructing rigged Hilbert spaces using time-delay embedding, significantly extending the algorithm's applicability (Rigged DMD can be used with any rigging). We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity, demonstrating Rigged DMD's convergence, efficiency, and versatility. This work paves the way for future research and applications of decompositions with continuous spectra.

Figures (13)

• Figure 1: Schematic of Rigged DMD. Stage A consists of a data-driven unitary approximation of $\mathcal{K}$. Stage B consists of sampling the resolvent of this approximation to form a smoothed generalized Koopman eigenfunction (and Koopman modes) in the form of a wave-packet approximation. These generalized eigenfunctions provide a decomposition, even in the presence of continuous spectra. The full algorithm is given in \ref{['alg:RiggedDMD']}. \newlabelfig:rigged_DMD0
• Figure 1: The essence of Stone's formula. Left: The points $z$ marked with $\bullet$ are where we evaluate $(\mathcal{K}-zI)^{-1}$, at a distance $\epsilon$ and $1-1/(1+\epsilon)$ from the unit circle. Right: The Poisson kernel for the unit disc. As $\epsilon\downarrow 0$, the Poisson kernel converges in the sense of distributions to a Dirac delta distribution.
• Figure 1: Left: The density of the spectral measure $\xi_g$ for the cat map example. Middle: The $L^\infty({[{-}\pi,\pi]_{\mathrm{per}}})$ error of the smoothed approximation computed using the kernels in \ref{['fig:kernels']}. Right: The error of the smoothed generalized eigenfunctions of the cat map computed using \ref{['alg:RiggedDMD']}. \newlabelfig:catmap10
• Figure 2: Spectral measure and generalized eigenfunctions for the Lorenz system computed using Rigged DMD. The spectral measure is continuous, and the generalized eigenfunctions are coherent. \newlabelfig:lorenz10
• Figure 2: The $m$th order kernels on ${[{-}\pi,\pi]_{\mathrm{per}}}$ constructed via periodic summation of rational kernels with equispaced poles in $[-1,1]$. They localize around the origin as $\epsilon\downarrow 0$, approximating Dirac's delta measure. \newlabelfig:kernels0

Theorems & Definitions (33)

• Example 3.1: Non-convergence of EDMD
• Theorem 4.1
• Proof 1
• Theorem 5.1
• Proof 2
• Definition 5.2: $m$th order periodic kernel
• Definition 5.3: $m$th order kernel for $\mathbb{R}$
• Proposition 5.4: Periodic summation of kernels
• Proof 3
• Proposition 5.5