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Modularized data-driven approximation of the Koopman operator and generator

Yang Guo, Manuel Schaller, Karl Worthmann, Stefan Streif

TL;DR

The paper addresses scalable data-driven Koopman analysis for interconnected nonlinear systems by introducing modularized generator EDMD (mgEDMD), which learns local Koopman generators on subsystems and then couples them according to the network topology. It derives finite-data error bounds for acyclic, weakly connected, and cyclic digraphs using graph-theoretic arguments, enabling provable performance guarantees. Numerical experiments on coupled Duffing and Van-der-Pol oscillators demonstrate data efficiency and clear advantages over holistic EDMD variants, while also enabling transfer learning to topology changes and identical/partially identical subsystems. The work offers a scalable, transfer-friendly framework for data-driven analysis, prediction, and control of large-scale nonlinear networks, with practical impact in modular modeling and distributed control.

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a widely-used data-driven approach to learn an approximation of the Koopman operator. Consequently, it provides a powerful tool for data-driven analysis, prediction, and control of nonlinear dynamical (control) systems. In this work, we propose a novel modularized EDMD scheme tailored to interconnected systems. To this end, we utilize the structure of the Koopman generator that allows to learn the dynamics of subsystems individually and thus alleviates the curse of dimensionality by considering observable functions on smaller state spaces. Moreover, our approach canonically enables transfer learning if a system encompasses multiple copies of a model as well as efficient adaption to topology changes without retraining. We provide finite-data bounds on the estimation error using tools from graph theory. The efficacy of the method is illustrated by means of various numerical examples.

Modularized data-driven approximation of the Koopman operator and generator

TL;DR

The paper addresses scalable data-driven Koopman analysis for interconnected nonlinear systems by introducing modularized generator EDMD (mgEDMD), which learns local Koopman generators on subsystems and then couples them according to the network topology. It derives finite-data error bounds for acyclic, weakly connected, and cyclic digraphs using graph-theoretic arguments, enabling provable performance guarantees. Numerical experiments on coupled Duffing and Van-der-Pol oscillators demonstrate data efficiency and clear advantages over holistic EDMD variants, while also enabling transfer learning to topology changes and identical/partially identical subsystems. The work offers a scalable, transfer-friendly framework for data-driven analysis, prediction, and control of large-scale nonlinear networks, with practical impact in modular modeling and distributed control.

Abstract

Extended Dynamic Mode Decomposition (EDMD) is a widely-used data-driven approach to learn an approximation of the Koopman operator. Consequently, it provides a powerful tool for data-driven analysis, prediction, and control of nonlinear dynamical (control) systems. In this work, we propose a novel modularized EDMD scheme tailored to interconnected systems. To this end, we utilize the structure of the Koopman generator that allows to learn the dynamics of subsystems individually and thus alleviates the curse of dimensionality by considering observable functions on smaller state spaces. Moreover, our approach canonically enables transfer learning if a system encompasses multiple copies of a model as well as efficient adaption to topology changes without retraining. We provide finite-data bounds on the estimation error using tools from graph theory. The efficacy of the method is illustrated by means of various numerical examples.
Paper Structure (23 sections, 8 theorems, 71 equations, 11 figures, 1 algorithm)

This paper contains 23 sections, 8 theorems, 71 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution
  3. Organization of the article
  4. Nomenclature
  5. Koopman-based representation and data-driven approximation
  6. The Koopman framework
  7. Data-driven approximation of the Koopman generator by EDMD
  8. A novel EDMD architecture for interconnected models
  9. Learning interconnected models via modularized generator EDMD
  10. Modularized EDMD using the Koopman operator
  11. Finite-data error bounds for modulized generator EDMD
  12. Error bounds for systems on acyclic digraphs
  13. Error bounds for weakly interconnected systems on arbitary digraphs
  14. Error bounds for systems on cyclic digraphs
  15. Numerical simulations for data-efficiency
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let the data points $\{x^l\}_{l\in [0:m-1]}$ be drawn independently and identically distributed (i.i.d.) w.r.t. the uniform distribution in $\mathbb X$. Then, for any error bound $\varepsilon >0$ and probabilistic tolerance $\delta \in (0,1)$ there is a sufficient amount of data $m_0 = \mathcal{O}(\

Figures (11)

  • Figure 1: Condensation of a cyclic diagraph.
  • Figure 2: Interconnected Duffing oscillators
  • Figure 3: Prediction errors for Duffing oscillators with different approaches
  • Figure 4: Interconnected Van-der-Pol oscillators
  • Figure 5: Prediction errors for Van-der-Pol oscillators
  • ...and 6 more figures

Theorems & Definitions (16)

  • Lemma 2.1
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.4
  • proof
  • Proposition 4.6
  • proof
  • ...and 6 more