A Low Rank Neural Representation of Entropy Solutions
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914255161262080 |
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| author | Rim, Donsub Welper, Gerrit |
| author_facet | Rim, Donsub Welper, Gerrit |
| contents | We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and posseses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low rank structure. Finally, we show that the low rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05694 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Low Rank Neural Representation of Entropy Solutions Rim, Donsub Welper, Gerrit Numerical Analysis Machine Learning 68T07, 41A46, 41A25, 65N15, 35L65 We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and posseses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low rank structure. Finally, we show that the low rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics. |
| title | A Low Rank Neural Representation of Entropy Solutions |
| topic | Numerical Analysis Machine Learning 68T07, 41A46, 41A25, 65N15, 35L65 |
| url | https://arxiv.org/abs/2406.05694 |