Efficient PINNs via multi-head unimodular regularization of the solutions space
Abstract Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differ...
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2025-08-01
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| Series: | Communications Physics |
| Online Access: | https://doi.org/10.1038/s42005-025-02248-1 |
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| author | Pedro Tarancón-Álvarez Pablo Tejerina-Pérez Raul Jimenez Pavlos Protopapas |
| author_facet | Pedro Tarancón-Álvarez Pablo Tejerina-Pérez Raul Jimenez Pavlos Protopapas |
| author_sort | Pedro Tarancón-Álvarez |
| collection | DOAJ |
| description | Abstract Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics-Informed Neural Networks (PINNs). This framework is based on what is called multi-head (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multi-head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations. |
| format | Article |
| id | doaj-art-57079ff6d2b246df9f73a09040365b26 |
| institution | Kabale University |
| issn | 2399-3650 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Communications Physics |
| spelling | doaj-art-57079ff6d2b246df9f73a09040365b262025-08-20T04:02:57ZengNature PortfolioCommunications Physics2399-36502025-08-018111410.1038/s42005-025-02248-1Efficient PINNs via multi-head unimodular regularization of the solutions spacePedro Tarancón-Álvarez0Pablo Tejerina-Pérez1Raul Jimenez2Pavlos Protopapas3Departament de Física Quántica i Astrofísica, Universitat de BarcelonaDepartament de Física Quántica i Astrofísica, Universitat de BarcelonaInstitut de Ciències del Cosmos (ICC), Universitat de BarcelonaInstitute for Applied Computational Science, Harvard UniversityAbstract Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics-Informed Neural Networks (PINNs). This framework is based on what is called multi-head (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multi-head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations.https://doi.org/10.1038/s42005-025-02248-1 |
| spellingShingle | Pedro Tarancón-Álvarez Pablo Tejerina-Pérez Raul Jimenez Pavlos Protopapas Efficient PINNs via multi-head unimodular regularization of the solutions space Communications Physics |
| title | Efficient PINNs via multi-head unimodular regularization of the solutions space |
| title_full | Efficient PINNs via multi-head unimodular regularization of the solutions space |
| title_fullStr | Efficient PINNs via multi-head unimodular regularization of the solutions space |
| title_full_unstemmed | Efficient PINNs via multi-head unimodular regularization of the solutions space |
| title_short | Efficient PINNs via multi-head unimodular regularization of the solutions space |
| title_sort | efficient pinns via multi head unimodular regularization of the solutions space |
| url | https://doi.org/10.1038/s42005-025-02248-1 |
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