Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers
In this work, we consider the time-harmonic Maxwell's equations and their numerical solution with a domain decomposition method. As an innovative feature, we propose a feedforward neural network-enhanced approximation of the interface conditions between the subdomains. The advantage is that the...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | Spanish |
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Universidad Nacional de Trujillo
2023-06-01
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| Series: | Selecciones Matemáticas |
| Subjects: | |
| Online Access: | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/5045 |
| _version_ | 1797802291334479872 |
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| author | Tobias Knoke Sebastian Kinnewig Sven Beuchler Ayhan Demircan Uwe Morgner Thomas Wick |
| author_facet | Tobias Knoke Sebastian Kinnewig Sven Beuchler Ayhan Demircan Uwe Morgner Thomas Wick |
| author_sort | Tobias Knoke |
| collection | DOAJ |
| description | In this work, we consider the time-harmonic Maxwell's equations and their numerical solution with a domain decomposition method. As an innovative feature, we propose a feedforward neural network-enhanced approximation of the interface conditions between the subdomains. The advantage is that the interface condition can be updated without recomputing the Maxwell system at each step. The main part consists of a detailed description of the construction of the neural network for domain decomposition and the training process. To substantiate this proof of concept, we investigate a few subdomains in some numerical experiments with low frequencies. Therein the new approach is compared to a classical domain decomposition method. Moreover, we highlight current challenges of training and testing with different wave numbers and we provide information on the behaviour of the neural-network, such as convergence of the loss function, and different activation functions. |
| first_indexed | 2024-03-13T05:03:29Z |
| format | Article |
| id | doaj.art-208ee1805aeb41a3be6ef2cf6e0092b5 |
| institution | Directory Open Access Journal |
| issn | 2411-1783 |
| language | Spanish |
| last_indexed | 2024-03-13T05:03:29Z |
| publishDate | 2023-06-01 |
| publisher | Universidad Nacional de Trujillo |
| record_format | Article |
| series | Selecciones Matemáticas |
| spelling | doaj.art-208ee1805aeb41a3be6ef2cf6e0092b52023-06-17T03:31:39ZspaUniversidad Nacional de TrujilloSelecciones Matemáticas2411-17832023-06-01100111510.17268/sel.mat.2023.01.01Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbersTobias Knoke0https://orcid.org/0000-0003-2987-5110Sebastian Kinnewig1https://orcid.org/0000-0002-0923-7413Sven Beuchler2https://orcid.org/0000-0001-9411-8701Ayhan Demircan3https://orcid.org/0000-0002-0015-2077Uwe Morgner4https://orcid.org/0000-0001-5103-9632Thomas Wick5https://orcid.org/0000-0002-1102-6332Institute of Applied Mathematics at Leibniz University Hannover, GermanyInstitute of Applied Mathematics at Leibniz University Hannover, GermanyInstitute of Applied Mathematics at Leibniz University Hannover, GermanyInstitute of Quantum Optics at Leibniz University Hannover, GermanyInstitute of Quantum Optics at Leibniz University Hannover, GermanyInstitute of Applied Mathematics at Leibniz University Hannover, GermanyIn this work, we consider the time-harmonic Maxwell's equations and their numerical solution with a domain decomposition method. As an innovative feature, we propose a feedforward neural network-enhanced approximation of the interface conditions between the subdomains. The advantage is that the interface condition can be updated without recomputing the Maxwell system at each step. The main part consists of a detailed description of the construction of the neural network for domain decomposition and the training process. To substantiate this proof of concept, we investigate a few subdomains in some numerical experiments with low frequencies. Therein the new approach is compared to a classical domain decomposition method. Moreover, we highlight current challenges of training and testing with different wave numbers and we provide information on the behaviour of the neural-network, such as convergence of the loss function, and different activation functions.https://revistas.unitru.edu.pe/index.php/SSMM/article/view/5045time-harmonic maxwell's equationsmachine learningfeedforward neural networkdomain decomposition method |
| spellingShingle | Tobias Knoke Sebastian Kinnewig Sven Beuchler Ayhan Demircan Uwe Morgner Thomas Wick Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers Selecciones Matemáticas time-harmonic maxwell's equations machine learning feedforward neural network domain decomposition method |
| title | Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers |
| title_full | Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers |
| title_fullStr | Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers |
| title_full_unstemmed | Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers |
| title_short | Domain Decomposition with Neural Network Interface Approximations for time-harmonic Maxwell's equations with different wave numbers |
| title_sort | domain decomposition with neural network interface approximations for time harmonic maxwell s equations with different wave numbers |
| topic | time-harmonic maxwell's equations machine learning feedforward neural network domain decomposition method |
| url | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/5045 |
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