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Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

Yuka Hashimoto, Tomoharu Iwata

TL;DR

The paper addresses the challenge of analyzing nonlinear time-series, especially nonautonomous dynamics, by embedding Koopman operator theory within a deep, layered architecture. It builds a Fourier-basis approximation space on $\mathbb{T}^d$ and replaces classical dictionaries with learnable Toeplitz-generators, forming multiple Koopman layers that are efficiently computed via Krylov subspace methods. The authors establish universality and RKHS-based generalization bounds for the framework, and demonstrate improved eigenvalue estimation for nonautonomous systems as well as enhanced forecasting on real-world time-series datasets. By connecting Koopman operator theory with numerical linear algebra, the approach offers a scalable, theoretically grounded alternative to purely neural or EDMD-based methods with practical impact on time-series analysis and forecasting.

Abstract

We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlying the model, we show its universality and generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations, which establish a new connection between Koopman operators and numerical linear algebra. We also empirically demonstrate that the proposed model outperforms existing methods on eigenvalue estimation of multiple Koopman operators for nonautonomous systems.

Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

TL;DR

The paper addresses the challenge of analyzing nonlinear time-series, especially nonautonomous dynamics, by embedding Koopman operator theory within a deep, layered architecture. It builds a Fourier-basis approximation space on and replaces classical dictionaries with learnable Toeplitz-generators, forming multiple Koopman layers that are efficiently computed via Krylov subspace methods. The authors establish universality and RKHS-based generalization bounds for the framework, and demonstrate improved eigenvalue estimation for nonautonomous systems as well as enhanced forecasting on real-world time-series datasets. By connecting Koopman operator theory with numerical linear algebra, the approach offers a scalable, theoretically grounded alternative to purely neural or EDMD-based methods with practical impact on time-series analysis and forecasting.

Abstract

We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlying the model, we show its universality and generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations, which establish a new connection between Koopman operators and numerical linear algebra. We also empirically demonstrate that the proposed model outperforms existing methods on eigenvalue estimation of multiple Koopman operators for nonautonomous systems.
Paper Structure (40 sections, 9 theorems, 15 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 40 sections, 9 theorems, 15 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Koopman generator and operator
  5. Deep Koopman-layered model
  6. Multiple dynamical systems and Koopman generators
  7. Approximation of Koopman generators using Toeplitz matrices
  8. Practical implementation
  9. Universality
  10. Generalization bound
  11. Numerical results
  12. Representation power and generalization
  13. Eigenvalues of the Koopman-layers for nonautonomous systems
  14. Measure-preserving dynamical system
  15. Damping oscillator with external force
  16. ...and 25 more sections

Key Result

Proposition 3

The $(n,l)$-entry of the representation matrix $Q_N^*LQ_N$ is where $l_k$ is the $k$th element of the multi index $l\in\mathbb{Z}^d$. Moreover, we set $n_r=m_{R_j}+\cdots +m_r+l$, thus $n_1=n$, $m_r=n_{r}-n_{r+1}$ for $r=1,\ldots,R-1$, and $m_{R}=n_{R}-l$.

Figures (12)

  • Figure 1: Overview of the deep Koopman-layered model.
  • Figure 2: Test errors for different values of $J$ with and without the regularization based on the norms of the Koopman operators. (The average $\pm$ standard deviation of three independent runs.)
  • Figure 3: Eigenvalues of the estimated Koopman operators for the nonautonomous measure preserving system.
  • Figure 4: Eigenvalues of the estimated Koopman operators for the nonautonomous damping oscillator. Full results are in Appendix \ref{['ap:exp_eig_addition']}.
  • Figure 5: Overview of the computation of the prediction $\hat{x}_t$ using $x_{t-2}$ and $x_{t-1}$.
  • ...and 7 more figures

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Proposition 3
  • Remark 4
  • Remark 5
  • Theorem 6
  • Corollary 7
  • Remark 8
  • Remark 9
  • Proposition 10
  • ...and 7 more