Approximation of nearly-periodic symplectic maps via structure-preserving neural networks
Abstract A continuous-time dynamical system with parameter $$\varepsilon$$ ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as $$\varepsilon$$ ε approaches 0. Nearly-periodic maps are discrete-time analogues of nearly-periodic systems, defined as par...
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2023-05-01
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| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-023-34862-w |
| _version_ | 1827895158892396544 |
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| author | Valentin Duruisseaux Joshua W. Burby Qi Tang |
| author_facet | Valentin Duruisseaux Joshua W. Burby Qi Tang |
| author_sort | Valentin Duruisseaux |
| collection | DOAJ |
| description | Abstract A continuous-time dynamical system with parameter $$\varepsilon$$ ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as $$\varepsilon$$ ε 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. |
| first_indexed | 2024-03-12T22:17:31Z |
| format | Article |
| id | doaj.art-158e1a4e2b5740b58e6a7a94d5281c3b |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-03-12T22:17:31Z |
| publishDate | 2023-05-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-158e1a4e2b5740b58e6a7a94d5281c3b2023-07-23T11:14:19ZengNature PortfolioScientific Reports2045-23222023-05-0113111910.1038/s41598-023-34862-wApproximation of nearly-periodic symplectic maps via structure-preserving neural networksValentin Duruisseaux0Joshua W. Burby1Qi Tang2Department of Mathematics, University of California San DiegoTheoretical Division, Los Alamos National LaboratoryTheoretical Division, Los Alamos National LaboratoryAbstract A continuous-time dynamical system with parameter $$\varepsilon$$ ε is nearly-periodic if all its trajectories are periodic with nowhere-vanishing angular frequency as $$\varepsilon$$ ε 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.https://doi.org/10.1038/s41598-023-34862-w |
| spellingShingle | Valentin Duruisseaux Joshua W. Burby Qi Tang Approximation of nearly-periodic symplectic maps via structure-preserving neural networks Scientific Reports |
| 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 |
| url | https://doi.org/10.1038/s41598-023-34862-w |
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