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Dynamic Mode Decomposition with Non-uniform Sampling

Ramachandran Anantharaman, Alexandre Mauroy

TL;DR

This work extends Dynamic Mode Decomposition and EDMD to settings where full state measurements are unavailable or non-uniform in time across components. The authors introduce a two-step algorithm: first, perform Hankel DMD on each state component to estimate its value at specified times separated by $T_s$, and second, apply (E)DMD on the reconstructed state to obtain lifted dynamics via the Koopman operator. They explore two practical instantiations—multirate EDMD and single-state EDMD—demonstrating on the Lorenz system that the proposed method yields more accurate Koopman spectra and trajectory predictions than naive approaches and can approach the performance of EDMD with complete data. The approach broadens the applicability of data-driven Koopman analysis to partially observed, non-uniform sampling scenarios typical of real sensing systems, with potential extensions to non-autonomous dynamics and theoretical guarantees.

Abstract

Dynamic Mode Decomposition (DMD) and its extensions (EDMD) have been at the forefront of data-based approaches to Koopman operators. Most (E)DMD algorithms assume that the entire state is sampled at a uniform sampling rate. In this paper, we provide an algorithm where the entire state is not uniformly sampled, with individual components of the states measured at individual (but known) sampling rates. We propose a two-step DMD algorithm where the first step performs Hankel DMD on individual state components to estimate them at specified time instants. With the entire state reconstructed at the same time instants, we compute the (E)DMD for the system with the estimated data in the second step.

Dynamic Mode Decomposition with Non-uniform Sampling

TL;DR

This work extends Dynamic Mode Decomposition and EDMD to settings where full state measurements are unavailable or non-uniform in time across components. The authors introduce a two-step algorithm: first, perform Hankel DMD on each state component to estimate its value at specified times separated by , and second, apply (E)DMD on the reconstructed state to obtain lifted dynamics via the Koopman operator. They explore two practical instantiations—multirate EDMD and single-state EDMD—demonstrating on the Lorenz system that the proposed method yields more accurate Koopman spectra and trajectory predictions than naive approaches and can approach the performance of EDMD with complete data. The approach broadens the applicability of data-driven Koopman analysis to partially observed, non-uniform sampling scenarios typical of real sensing systems, with potential extensions to non-autonomous dynamics and theoretical guarantees.

Abstract

Dynamic Mode Decomposition (DMD) and its extensions (EDMD) have been at the forefront of data-based approaches to Koopman operators. Most (E)DMD algorithms assume that the entire state is sampled at a uniform sampling rate. In this paper, we provide an algorithm where the entire state is not uniformly sampled, with individual components of the states measured at individual (but known) sampling rates. We propose a two-step DMD algorithm where the first step performs Hankel DMD on individual state components to estimate them at specified time instants. With the entire state reconstructed at the same time instants, we compute the (E)DMD for the system with the estimated data in the second step.
Paper Structure (14 sections, 42 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 42 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Data-based approaches to Koopman operator
  4. Implementation of DMD and computing lifted dynamics
  5. Hankel DMD
  6. EDMD with non-uniform sampling of states
  7. Estimate of the states through Hankel DMD
  8. EDMD with estimated data
  9. Applications of EDMD with non-uniform sampling
  10. Multirate Sampling
  11. Numerical Example
  12. Single State Sampling
  13. Numerical Example
  14. Conclusion

Figures (4)

  • Figure 1: Spectrum of the $\mathbf{L}_N$ matrix: It can be seen that the spectrum of the infinitesimal generator using multirate EDMD with estimated data better approximates the EDMD spectrum with complete measurements at $x(T_s)$ than the spectrum computed with the data $x(MT_s)$
  • Figure 2: Time prediction with EDMD: The trajectories are better predicted with the proposed multi-rate EDMD than the EDMD performed with data at $x(MT_s)$
  • Figure 3: Spectrum of the $\mathbf{L}_N$ matrix: Comparision of the spectrum of EDMD performed with complete data at $T_s$ and proposed single state EDMD
  • Figure 4: Time prediction with EDMD

Theorems & Definitions (1)

  • Remark 1