Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods
This paper introduces an innovative methodology for solving steady-state elliptic partial differential equations defined over irregular domains, by coupling the capabilities of Physics-Informed Neural Networks with the Fictitious Domain Method. The primary emphasis is placed on applications involvin...
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| Format: | Article |
| Language: | English |
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Taylor & Francis
2025-12-01
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| Series: | Research in Statistics |
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| Online Access: | https://www.tandfonline.com/doi/10.1080/27684520.2025.2542577 |
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| author | José A. Rodrigues |
| author_facet | José A. Rodrigues |
| author_sort | José A. Rodrigues |
| collection | DOAJ |
| description | This paper introduces an innovative methodology for solving steady-state elliptic partial differential equations defined over irregular domains, by coupling the capabilities of Physics-Informed Neural Networks with the Fictitious Domain Method. The primary emphasis is placed on applications involving the heat equation, a fundamental model in thermal analysis where the complexity of non-standard geometries often poses significant challenges for traditional numerical methods. The proposed approach exploits the inherent strength of Physics-Informed Neural Networks in embedding the underlying physical laws directly into the learning process, enabling the model to approximate solutions without relying on mesh-based discretization. Simultaneously, the Fictitious Domain Method facilitates the treatment of irregular computational domains by embedding them within a larger, regular domain, thereby simplifying the application of boundary conditions and numerical operations. The synergy between these two techniques results in a flexible, efficient, and accurate computational framework that is well-suited for addressing heat transfer problems in complex geometrical configurations. |
| format | Article |
| id | doaj-art-ae815e34f4064e069bbcac66791a0997 |
| institution | Kabale University |
| issn | 2768-4520 |
| language | English |
| publishDate | 2025-12-01 |
| publisher | Taylor & Francis |
| record_format | Article |
| series | Research in Statistics |
| spelling | doaj-art-ae815e34f4064e069bbcac66791a09972025-08-25T13:01:35ZengTaylor & FrancisResearch in Statistics2768-45202025-12-013110.1080/27684520.2025.2542577Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain MethodsJosé A. Rodrigues0CIMA and Department of Mathematics of ISEL, Polytechnic University of Lisbon, PortugalThis paper introduces an innovative methodology for solving steady-state elliptic partial differential equations defined over irregular domains, by coupling the capabilities of Physics-Informed Neural Networks with the Fictitious Domain Method. The primary emphasis is placed on applications involving the heat equation, a fundamental model in thermal analysis where the complexity of non-standard geometries often poses significant challenges for traditional numerical methods. The proposed approach exploits the inherent strength of Physics-Informed Neural Networks in embedding the underlying physical laws directly into the learning process, enabling the model to approximate solutions without relying on mesh-based discretization. Simultaneously, the Fictitious Domain Method facilitates the treatment of irregular computational domains by embedding them within a larger, regular domain, thereby simplifying the application of boundary conditions and numerical operations. The synergy between these two techniques results in a flexible, efficient, and accurate computational framework that is well-suited for addressing heat transfer problems in complex geometrical configurations.https://www.tandfonline.com/doi/10.1080/27684520.2025.2542577Physics-Informed Neural NetworksFictitious Domain MethodIrregular DomainsSteady-State Heat ConductionNURBS Curves |
| spellingShingle | José A. Rodrigues Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods Research in Statistics Physics-Informed Neural Networks Fictitious Domain Method Irregular Domains Steady-State Heat Conduction NURBS Curves |
| title | Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods |
| title_full | Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods |
| title_fullStr | Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods |
| title_full_unstemmed | Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods |
| title_short | Solving Steady-State Elliptic Problems in Irregular Domains Using Physics-Informed Neural Networks and Fictitious Domain Methods |
| title_sort | solving steady state elliptic problems in irregular domains using physics informed neural networks and fictitious domain methods |
| topic | Physics-Informed Neural Networks Fictitious Domain Method Irregular Domains Steady-State Heat Conduction NURBS Curves |
| url | https://www.tandfonline.com/doi/10.1080/27684520.2025.2542577 |
| work_keys_str_mv | AT josearodrigues solvingsteadystateellipticproblemsinirregulardomainsusingphysicsinformedneuralnetworksandfictitiousdomainmethods |