Extended dynamic mode decomposition with Fourier dictionaries: Error bounds and fast implementation
Abstract
The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we propose an EDMD-variant with a Fourier dictionary on the d-dimensional torus, where the data are sampled on an equispaced tensor grid. We show that the EDMD regression problem admits a unique closed-form solution and coincides with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive Lp-error bounds, 2 ≤ p ≤ ∞, on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation can be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators and for the chaotic ABC flow with dictionaries of up to 107 modes.
Related
- pqSEDMD: Subspace Methods for Extended Dynamic Mode Decomposition Identification
- Fourier Decomposition and E-Eigenvalues of Hypergraph Polynomials over Finite Abelian Groups
- Functional dynamic mode decomposition: Learning infinite-dimensional systems from data
- High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics
- Regime-Aware Estimation of Large Equity Correlation Matrices: Tail-Pairwise Dependence and Dynamic Conditional Correlation Decomposition for Water and Energy Sectors
BibTeX
@misc{bartel2026extended,
title = {{Extended dynamic mode decomposition with Fourier dictionaries: Error bounds and fast implementation}},
author = {Felix Bartel and Sandra Ritter and Manuel Schaller and Karl Worthmann},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20144},
eprint = {2609.20144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20144},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Extended dynamic mode decomposition with Fourier dictionaries: Error bounds and fast implementation
AU - Felix Bartel
AU - Sandra Ritter
AU - Manuel Schaller
AU - Karl Worthmann
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2609.20144
UR - https://arxiv.org/abs/2609.20144
AB - The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we propose an EDMD-variant with a Fourier dictionary on the d-dimensional torus, where the data are sampled on an equispaced tensor grid. We show that the EDMD regression problem admits a unique closed-form solution and coincides with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive L^p-error bounds, 2 ≤ p ≤ ∞, on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation can be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators and for the chaotic ABC flow with dictionaries of up to 10^7 modes.
ER -