Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913039938224128 |
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| author | Hertel, Maximiliano Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl |
| author_facet | Hertel, Maximiliano Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl |
| contents | We prove $L^\infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems. To this end, we establish Koopman invariance of suitably chosen reproducing kernel Hilbert spaces and provide an in-depth analysis of the pointwise error in terms of the data points. The latter is split into two parts by showing that kEDMD for stochastic systems involves a kernel regression step leading to a deterministic error in the fill distance as well as Monte Carlo sampling to approximate unknown expected values yielding a probabilistic error in terms of the number of samples. We illustrate the derived bounds by means of Langevin-type stochastic differential equations involving a nonlinear double-well potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition Hertel, Maximiliano Philipp, Friedrich M. Schaller, Manuel Worthmann, Karl Dynamical Systems Numerical Analysis 37M99, 47B32, 65C05, 65D12 We prove $L^\infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems. To this end, we establish Koopman invariance of suitably chosen reproducing kernel Hilbert spaces and provide an in-depth analysis of the pointwise error in terms of the data points. The latter is split into two parts by showing that kEDMD for stochastic systems involves a kernel regression step leading to a deterministic error in the fill distance as well as Monte Carlo sampling to approximate unknown expected values yielding a probabilistic error in terms of the number of samples. We illustrate the derived bounds by means of Langevin-type stochastic differential equations involving a nonlinear double-well potential. |
| title | Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition |
| topic | Dynamical Systems Numerical Analysis 37M99, 47B32, 65C05, 65D12 |
| url | https://arxiv.org/abs/2512.20247 |