[Paper Review] Latent Modes of Nonlinear Flows -- a Koopman Theory Analysis

I. Bernard Cohen, Guy Gilboa|arXiv (Cornell University)|Jul 15, 2021

Model Reduction and Neural Networks37 references4 citations

TL;DR

This paper introduces a Koopman theory-based framework for analyzing nonlinear flows by identifying latent structures through Koopman Eigenfunctions (KEFs), enabling global linearization, dimensionality reduction, and system reconstruction. It proposes a novel sparsity-driven mode decomposition using an overcomplete dictionary of decay profiles, overcoming limitations of traditional Dynamic Mode Decomposition (DMD) in capturing complex dynamics.

ABSTRACT

Extracting the latent underlying structures of complex nonlinear local and nonlocal flows is essential for their analysis and modeling. In this work, we attempt to provide a consistent framework through Koopman theory and its related popular discrete approximation -- dynamic mode decomposition (DMD). We investigate the conditions to perform appropriate linearization, dimensionality reduction, and representation of flows in a highly general setting. The essential elements of this framework are Koopman Eigenfunction (KEF), for which existence conditions are formulated. This is done by viewing the dynamic as a curve in state-space. These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics. We examine the limitations of DMD through the analysis of Koopman theory and propose a new mode decomposition technique based on the typical time profile of the dynamics. An overcomplete dictionary of decay profiles is used to sparsely approximate the flow. This analysis is also valid in the full continuous setting of Koopman theory, which is based on variational calculus. We demonstrate applications of this analysis, such as finding KEFs and their multiplicities, dynamics reconstruction, and global linearization.

Motivation & Objective

• To develop a consistent framework for analyzing complex nonlinear flows using Koopman theory and Dynamic Mode Decomposition (DMD).
• To formulate existence conditions for Koopman Eigenfunctions (KEFs) in a general nonlinear setting.
• To address the limitations of DMD in representing nonlinear dynamics through a novel mode decomposition technique.
• To enable system reconstruction, global controllability, and observability for nonlinear systems via KEFs.
• To extend the analysis to the continuous setting using variational calculus and overcomplete dictionaries of decay profiles.

Proposed method

• Model the dynamic as a curve in state-space to analyze Koopman operator properties and derive existence conditions for KEFs.
• Use an overcomplete dictionary of typical time profiles (e.g., exponential decay) to sparsely approximate the flow dynamics.
• Apply sparse approximation techniques to identify dominant modes from the dictionary, ensuring minimal representation error.
• Formulate the decomposition in both discrete (DMD-like) and continuous (variational calculus-based) settings.
• Integrate variational calculus to extend the framework to continuous-time Koopman theory.
• Reconstruct dynamics and assess global linearization by combining identified KEFs and their multiplicities.

Experimental results

Research questions

• RQ1 Under what conditions do Koopman Eigenfunctions (KEFs) exist for general nonlinear flows?
• RQ2 How can DMD be systematically improved to represent complex nonlinear dynamics beyond its standard assumptions?
• RQ3 What is the role of decay profile dictionaries in enabling sparse, accurate representation of nonlinear flow dynamics?
• RQ4 How can the continuous Koopman framework be extended using variational calculus for full dynamic analysis?
• RQ5 Can the identified KEFs and their multiplicities enable global system reconstruction and linearization?

Key findings

• The paper establishes sufficient conditions for the existence of Koopman Eigenfunctions (KEFs) in nonlinear flows through a state-space curve representation.
• The proposed sparsity-based decomposition outperforms standard DMD by capturing non-exponential and non-periodic dynamics through an overcomplete dictionary of decay profiles.
• KEFs and their multiplicities are successfully identified, enabling accurate reconstruction of nonlinear dynamics.
• The framework supports global linearization of nonlinear systems by embedding them into a higher-dimensional linear space via KEFs.
• The continuous Koopman theory extension using variational calculus enables rigorous analysis of flow dynamics beyond discrete approximations.
• The method demonstrates robustness in reconstructing complex flows, including gradient and homogeneous operator-based dynamics, with improved observability and controllability characterization.