Two-layers neural networks for Schr{ö}dinger eigenvalue problems
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929483961860096 |
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| author | Dus, Mathias Virginie, Ehrlacher |
| author_facet | Dus, Mathias Virginie, Ehrlacher |
| contents | The aim of this article is to analyze numerical schemes using two-layer neural networks withinfinite width for the resolution of high-dimensional Schr{ö}dinger eigenvalue problems with smoothinteraction potentials and Neumann boundary condition on the unit cube in any dimension. Moreprecisely, any eigenfunction associated to the lowest eigenvalue of the Schr{ö}dinger operator is a unitL 2 norm minimizer of the associated energy. Using Barron's representation of the solution witha probability measure defined on the set of parameter values and following the approach initiallysuggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curvedynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Weprove the existence of solutions to this constrained gradient curve. Furthermore, we prove that,if it converges, the represented function is then an eigenfunction of the considered Schr{ö}dingeroperator. At least up to our knowledge, this is the first work where this type of analysis is carriedout to deal with the minimization of non-convex functionals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01640 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-layers neural networks for Schr{ö}dinger eigenvalue problems Dus, Mathias Virginie, Ehrlacher Analysis of PDEs Optimization and Control Probability The aim of this article is to analyze numerical schemes using two-layer neural networks withinfinite width for the resolution of high-dimensional Schr{ö}dinger eigenvalue problems with smoothinteraction potentials and Neumann boundary condition on the unit cube in any dimension. Moreprecisely, any eigenfunction associated to the lowest eigenvalue of the Schr{ö}dinger operator is a unitL 2 norm minimizer of the associated energy. Using Barron's representation of the solution witha probability measure defined on the set of parameter values and following the approach initiallysuggested by Bach and Chizat [1], the energy is minimized thanks to a constrained gradient curvedynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Weprove the existence of solutions to this constrained gradient curve. Furthermore, we prove that,if it converges, the represented function is then an eigenfunction of the considered Schr{ö}dingeroperator. At least up to our knowledge, this is the first work where this type of analysis is carriedout to deal with the minimization of non-convex functionals. |
| title | Two-layers neural networks for Schr{ö}dinger eigenvalue problems |
| topic | Analysis of PDEs Optimization and Control Probability |
| url | https://arxiv.org/abs/2409.01640 |