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

Core Concepts

The Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm computes generalized eigenfunction decompositions of Koopman operators from snapshot data, enabling the analysis of dynamical systems with continuous spectra.

Abstract

The paper introduces the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. Koopman operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis, but 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. The algorithm builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition (mpEDMD) with high-order kernels for smoothing. This provides a robust decomposition encompassing both discrete and continuous spectral elements.
The authors derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures. They also propose a novel framework for constructing rigged Hilbert spaces using time-delay embedding, significantly extending the algorithm's applicability. The paper provides examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number fluid flow, demonstrating Rigged DMD's convergence, efficiency, and versatility.

Stats

"We derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures."
"The paper provides examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number fluid flow, demonstrating Rigged DMD's convergence, efficiency, and versatility."

Quotes

"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."
"The algorithm builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition (mpEDMD) with high-order kernels for smoothing."

Key Insights Distilled From

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

by Matthew J. C... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00782.pdf

Deeper Inquiries

How can the Rigged DMD framework be extended to handle non-unitary Koopman operators

To extend the Rigged DMD framework to handle non-unitary Koopman operators, we need to consider the generalization of the spectral theory to non-unitary operators. While the current framework focuses on unitary Koopman operators, the extension to non-unitary operators would involve adapting the decomposition and approximation techniques to accommodate the different spectral properties of non-unitary operators. This may require developing new algorithms and methodologies to compute the resolvent and generalized eigenfunctions for non-unitary Koopman operators. Additionally, the convergence theorems and regularization techniques would need to be redefined to suit the properties of non-unitary operators. By incorporating the characteristics of non-unitary operators into the Rigged DMD framework, we can effectively analyze and decompose the dynamics of a broader class of systems.

What are the limitations of the rigged Hilbert space construction via time-delay embedding, and how can it be further generalized

The rigged Hilbert space construction via time-delay embedding has certain limitations that can be addressed for further generalization. One limitation is the dependence on the choice of the time delay parameter in the embedding process. Different time delays can lead to varying results and may not capture the dynamics optimally. To overcome this limitation, techniques such as adaptive time delay selection or data-driven methods for determining the optimal time delay can be implemented. Another limitation is the assumption of linearity in the construction of the rigged Hilbert space. Extending the framework to handle nonlinear dynamics can enhance its applicability to a wider range of systems. By incorporating nonlinear transformations and adaptive strategies for time delay selection, the rigged Hilbert space construction can be further generalized to capture the complexities of nonlinear dynamical systems.

What are the potential applications of the generalized eigenfunction decompositions computed by Rigged DMD beyond the examples provided in the paper

The generalized eigenfunction decompositions computed by Rigged DMD have various potential applications beyond the examples provided in the paper. Some of these applications include:

Pattern Recognition: The coherent structures identified by the generalized eigenfunctions can be utilized for pattern recognition in complex systems. By analyzing the coherent modes, one can identify recurring patterns and behaviors within the system.
Anomaly Detection: The decomposition can be used for anomaly detection in dynamical systems. Deviations from the expected coherent modes can indicate anomalies or irregularities in the system's behavior.
Model Validation: The computed eigenfunctions can serve as a basis for validating and refining mathematical models of the system. Discrepancies between the model predictions and the observed generalized eigenfunctions can highlight areas for model improvement.
Control Strategies: The decompositions can aid in developing control strategies for dynamical systems. By understanding the coherent modes and their evolution, effective control mechanisms can be designed to steer the system towards desired states or behaviors.
These applications demonstrate the versatility and utility of generalized eigenfunction decompositions in various fields, ranging from engineering and physics to biology and finance.