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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.03426 |
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| _version_ | 1866911921080369152 |
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| author | Hillebrecht, Birgit Unger, Benjamin |
| author_facet | Hillebrecht, Birgit Unger, Benjamin |
| contents | Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03426 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs Hillebrecht, Birgit Unger, Benjamin Machine Learning Numerical Analysis Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation. |
| title | Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2210.03426 |