Estimatable variation neural networks and their application to scalar hyperbolic conservation laws
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911262575689728 |
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| author | Lukáčová-Medviďová, Mária Schneider, Simon |
| author_facet | Lukáčová-Medviďová, Mária Schneider, Simon |
| contents | We introduce estimatable variation neural networks (EVNNs), a class of neural networks that allow a computationally cheap estimate on the $BV$ norm motivated by the space $BMV$ of functions with bounded M-variation. We prove a universal approximation theorem for EVNNs and discuss possible implementations. We construct sequences of loss functionals for ODEs and scalar hyperbolic conservation laws for which a vanishing loss leads to convergence. Moreover, we show the existence of sequences of loss minimizing neural networks if the solution is an element of $BMV$. Several numerical test cases illustrate that it is possible to use standard techniques to minimize these loss functionals for EVNNs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_08909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Estimatable variation neural networks and their application to scalar hyperbolic conservation laws Lukáčová-Medviďová, Mária Schneider, Simon Numerical Analysis 65M99 (Primary), 35L65, 35L60, (Secondary) We introduce estimatable variation neural networks (EVNNs), a class of neural networks that allow a computationally cheap estimate on the $BV$ norm motivated by the space $BMV$ of functions with bounded M-variation. We prove a universal approximation theorem for EVNNs and discuss possible implementations. We construct sequences of loss functionals for ODEs and scalar hyperbolic conservation laws for which a vanishing loss leads to convergence. Moreover, we show the existence of sequences of loss minimizing neural networks if the solution is an element of $BMV$. Several numerical test cases illustrate that it is possible to use standard techniques to minimize these loss functionals for EVNNs. |
| title | Estimatable variation neural networks and their application to scalar hyperbolic conservation laws |
| topic | Numerical Analysis 65M99 (Primary), 35L65, 35L60, (Secondary) |
| url | https://arxiv.org/abs/2409.08909 |