Neural networks for the approximation of Euler's elastica
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arXiv
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| Hauptverfasser: | , , , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913375767756800 |
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| author | Celledoni, Elena Çokaj, Ergys Leone, Andrea Leyendecker, Sigrid Murari, Davide Owren, Brynjulf de Almagro, Rodrigo T. Sato Martín Stavole, Martina |
| author_facet | Celledoni, Elena Çokaj, Ergys Leone, Andrea Leyendecker, Sigrid Murari, Davide Owren, Brynjulf de Almagro, Rodrigo T. Sato Martín Stavole, Martina |
| contents | Euler's elastica is a classical model of flexible slender structures, relevant in many industrial applications. Static equilibrium equations can be derived via a variational principle. The accurate approximation of solutions of this problem can be challenging due to nonlinearity and constraints. We here present two neural network based approaches for the simulation of this Euler's elastica. Starting from a data set of solutions of the discretised static equilibria, we train the neural networks to produce solutions for unseen boundary conditions. We present a $\textit{discrete}$ approach learning discrete solutions from the discrete data. We then consider a $\textit{continuous}$ approach using the same training data set, but learning continuous solutions to the problem. We present numerical evidence that the proposed neural networks can effectively approximate configurations of the planar Euler's elastica for a range of different boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_00644 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Neural networks for the approximation of Euler's elastica Celledoni, Elena Çokaj, Ergys Leone, Andrea Leyendecker, Sigrid Murari, Davide Owren, Brynjulf de Almagro, Rodrigo T. Sato Martín Stavole, Martina Numerical Analysis Euler's elastica is a classical model of flexible slender structures, relevant in many industrial applications. Static equilibrium equations can be derived via a variational principle. The accurate approximation of solutions of this problem can be challenging due to nonlinearity and constraints. We here present two neural network based approaches for the simulation of this Euler's elastica. Starting from a data set of solutions of the discretised static equilibria, we train the neural networks to produce solutions for unseen boundary conditions. We present a $\textit{discrete}$ approach learning discrete solutions from the discrete data. We then consider a $\textit{continuous}$ approach using the same training data set, but learning continuous solutions to the problem. We present numerical evidence that the proposed neural networks can effectively approximate configurations of the planar Euler's elastica for a range of different boundary conditions. |
| title | Neural networks for the approximation of Euler's elastica |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2312.00644 |