Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918003891765248 |
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| author | Hauser, Till Hölz, Julian |
| author_facet | Hauser, Till Hölz, Julian |
| contents | Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace $F \subseteq \mathrm{L}^2(\mathrm{X})$ required for predicting the behavior of dynamical systems over long time horizons. We distinguish between two cases: (i) If $F$ is determined by a finite partition of $X$ we derive a lower bound that depends on the dynamical measure-theoretic entropy of the partition. (ii) We consider general finite-dimensional subspaces $F$ and establish a lower bound for the dimension of $F$ that is contingent on the spectral structure of the Koopman operator of the system, via the approximation entropy of $F$ as studied by Voiculescu. Furthermore, we motivate the use of delay observables to improve the predictive qualities of Dynamic Mode Decomposition algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms Hauser, Till Hölz, Julian Dynamical Systems Numerical Analysis Functional Analysis 37A05, 37M10, 37M25, 65P99 Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace $F \subseteq \mathrm{L}^2(\mathrm{X})$ required for predicting the behavior of dynamical systems over long time horizons. We distinguish between two cases: (i) If $F$ is determined by a finite partition of $X$ we derive a lower bound that depends on the dynamical measure-theoretic entropy of the partition. (ii) We consider general finite-dimensional subspaces $F$ and establish a lower bound for the dimension of $F$ that is contingent on the spectral structure of the Koopman operator of the system, via the approximation entropy of $F$ as studied by Voiculescu. Furthermore, we motivate the use of delay observables to improve the predictive qualities of Dynamic Mode Decomposition algorithms. |
| title | Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms |
| topic | Dynamical Systems Numerical Analysis Functional Analysis 37A05, 37M10, 37M25, 65P99 |
| url | https://arxiv.org/abs/2504.20269 |