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Functional Dimensionality of Koopman Eigenfunction Space

Ido Cohen, Eli Appleboim, Gershon Wolansky

TL;DR

The paper shows that the Koopman Partial Differential Equation has a general solution whose functional Koopman eigenfunction space is finite and whose dimensionality equals the underlying dynamics, enabling a perfect linear representation of nonlinear systems with only $N$ eigenfunctions. Using the characteristics method, it derives a KPDE solution and establishes a minimal set of eigenfunctions together with a flowbox transformation that linearizes the flow in a suitable coordinate system. It connects the minimal set concept, flowbox coordinates, and conservation laws, expressing eigenfunctions as $oldsymbol{ extPhi}(m{x})=f(h_1(m{x}), obreak\ldots,h_{N-1}(m{x}))e^{ obreak lambda M(m{x)}}$ and showing how the invariants drive the representation. The work provides a geometry-driven, finite-dimensional framework for stable, accurate representation and discovery of Koopman eigenfunctions from samples, with potential benefits for analysis and prediction in nonlinear dynamical systems.

Abstract

This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation.

Functional Dimensionality of Koopman Eigenfunction Space

TL;DR

The paper shows that the Koopman Partial Differential Equation has a general solution whose functional Koopman eigenfunction space is finite and whose dimensionality equals the underlying dynamics, enabling a perfect linear representation of nonlinear systems with only eigenfunctions. Using the characteristics method, it derives a KPDE solution and establishes a minimal set of eigenfunctions together with a flowbox transformation that linearizes the flow in a suitable coordinate system. It connects the minimal set concept, flowbox coordinates, and conservation laws, expressing eigenfunctions as and showing how the invariants drive the representation. The work provides a geometry-driven, finite-dimensional framework for stable, accurate representation and discovery of Koopman eigenfunctions from samples, with potential benefits for analysis and prediction in nonlinear dynamical systems.

Abstract

This work presents the general form solution of Koopman Partial Differential Equation and shows that its functional dimensionality is finite. The dimensionality is as the dimensionality of the dynamics. Thus, the representation of nonlinear dynamics as a linear one with a finite set of Koopman eigenfunctions without error is possible. This formulation justifies the flowbox statement and provides a simple numerical method to find such representation.
Paper Structure (10 sections, 3 theorems, 8 equations, 1 figure)

This paper contains 10 sections, 3 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Paper Outlines
  4. Preliminaries
  5. Solution of KPDE, Minimal Set, and Flowbox
  6. General Solution of KPDE
  7. Examples
  8. Minimal Set
  9. Flowbox
  10. Conclusions

Key Result

Theorem 3.1

Given a function $g$ on $S$, the equation has a unique solution in the domain $\Omega(S)$ satisfying $\Psi=g$ on $S$.

Figures (1)

  • Figure 1: Initial surfaces for a source and a hyperbolic systems

Theorems & Definitions (6)

  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4: General solution of KPDE
  • proof