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Approximation of nearly-periodic symplectic maps via structure-preserving neural networks
A continuous-time dynamical system with parameter [Formula: see text] is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as [Formula: see text] approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as paramete...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Nature Publishing Group UK
2023
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10206110/ https://www.ncbi.nlm.nih.gov/pubmed/37221253 http://dx.doi.org/10.1038/s41598-023-34862-w |
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author | Duruisseaux, Valentin Burby, Joshua W. Tang, Qi |
author_facet | Duruisseaux, Valentin Burby, Joshua W. Tang, Qi |
author_sort | Duruisseaux, Valentin |
collection | PubMed |
description | A continuous-time dynamical system with parameter [Formula: see text] is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as [Formula: see text] approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities. |
format | Online Article Text |
id | pubmed-10206110 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2023 |
publisher | Nature Publishing Group UK |
record_format | MEDLINE/PubMed |
spelling | pubmed-102061102023-05-25 Approximation of nearly-periodic symplectic maps via structure-preserving neural networks Duruisseaux, Valentin Burby, Joshua W. Tang, Qi Sci Rep Article A continuous-time dynamical system with parameter [Formula: see text] is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as [Formula: see text] approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as parameter-dependent diffeomorphisms that limit to rotations along a circle action, and they admit formal U(1) symmetries to all orders when the limiting rotation is non-resonant. For Hamiltonian nearly-periodic maps on exact presymplectic manifolds, the formal U(1) symmetry gives rise to a discrete-time adiabatic invariant. In this paper, we construct a novel structure-preserving neural network to approximate nearly-periodic symplectic maps. This neural network architecture, which we call symplectic gyroceptron, ensures that the resulting surrogate map is nearly-periodic and symplectic, and that it gives rise to a discrete-time adiabatic invariant and a long-time stability. This new structure-preserving neural network provides a promising architecture for surrogate modeling of non-dissipative dynamical systems that automatically steps over short timescales without introducing spurious instabilities. Nature Publishing Group UK 2023-05-23 /pmc/articles/PMC10206110/ /pubmed/37221253 http://dx.doi.org/10.1038/s41598-023-34862-w Text en © The Author(s) 2023, corrected publication 2023 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Duruisseaux, Valentin Burby, Joshua W. Tang, Qi Approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
title | Approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
title_full | Approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
title_fullStr | Approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
title_full_unstemmed | Approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
title_short | Approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
title_sort | approximation of nearly-periodic symplectic maps via structure-preserving neural networks |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10206110/ https://www.ncbi.nlm.nih.gov/pubmed/37221253 http://dx.doi.org/10.1038/s41598-023-34862-w |
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