Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential
We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-01-01
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| Online Access: | https://www.mdpi.com/1099-4300/23/1/95 |
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| author | Luka Grubišić Marko Hajba Domagoj Lacmanović |
| author_facet | Luka Grubišić Marko Hajba Domagoj Lacmanović |
| author_sort | Luka Grubišić |
| collection | DOAJ |
| description | We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we compute localized bounded states at the lower end of the spectrum. We present several deep network architectures that predict the localization of bounded states from a sample of a potential. For tackling higher dimensional problems, we consider a class of physics-informed deep dense networks. In particular, we focus on the interpretability of the proposed approaches. Deep network is used as a general reduced order model that describes the nonlinear connection between the potential and the ground state. The performance of the surrogate reduced model is controlled by an error estimator and the model is updated if necessary. Finally, we present a host of experiments to measure the accuracy and performance of the proposed algorithm. |
| first_indexed | 2024-03-09T05:15:40Z |
| format | Article |
| id | doaj.art-13622a54b67f417fb31c457bce19f50a |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T05:15:40Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-13622a54b67f417fb31c457bce19f50a2023-12-03T12:45:22ZengMDPI AGEntropy1099-43002021-01-012319510.3390/e23010095Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining PotentialLuka Grubišić0Marko Hajba1Domagoj Lacmanović2Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, CroatiaDepartment of ICT, Virovitica College, 33000 Virovitica, CroatiaDepartment of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, CroatiaWe study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials and study the associated effective confining potential. This problem is posed in the finite domain and we compute localized bounded states at the lower end of the spectrum. We present several deep network architectures that predict the localization of bounded states from a sample of a potential. For tackling higher dimensional problems, we consider a class of physics-informed deep dense networks. In particular, we focus on the interpretability of the proposed approaches. Deep network is used as a general reduced order model that describes the nonlinear connection between the potential and the ground state. The performance of the surrogate reduced model is controlled by an error estimator and the model is updated if necessary. Finally, we present a host of experiments to measure the accuracy and performance of the proposed algorithm.https://www.mdpi.com/1099-4300/23/1/95Anderson localizationdeep neural networksresidual error estimatesphysics informed neural networks |
| spellingShingle | Luka Grubišić Marko Hajba Domagoj Lacmanović Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential Entropy Anderson localization deep neural networks residual error estimates physics informed neural networks |
| title | Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential |
| title_full | Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential |
| title_fullStr | Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential |
| title_full_unstemmed | Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential |
| title_short | Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential |
| title_sort | deep neural network model for approximating eigenmodes localized by a confining potential |
| topic | Anderson localization deep neural networks residual error estimates physics informed neural networks |
| url | https://www.mdpi.com/1099-4300/23/1/95 |
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