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

arXiv: https://arxiv.org/abs/2405.00782 · HTML: https://mlsanity.com/paper/2405.00782/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 82 references that are in the map.
- First posted on arXiv 1 May 2024.

## Builds on (most cited first, 7 of 7)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [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)
- [The Multiverse of Dynamic Mode Decomposition Algorithms](https://mlsanity.com/paper/2312.00137/) (2023)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos](https://mlsanity.com/paper/2305.08074/) (2023)

## Built on by (most cited first, 7 of 7)

- [An Introductory Guide to Koopman Learning](https://mlsanity.com/paper/2510.22002/) (2025)
- [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)
- [Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework](https://mlsanity.com/paper/2605.14224/) (2026)
- [Avoiding spectral pollution for transfer operators using residuals](https://mlsanity.com/paper/2507.16915/) (2025)
- [Deep Embedded Multiplicative DMD for Algebra-Preserving Koopman Learning](https://mlsanity.com/paper/2606.05131/) (2026)

## Abstract (arXiv)

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.