A deep first-order system least squares method for the obstacle problem
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915465917366272 |
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| author | Acosta, Gabriel Belén, Eugenia Bersetche, Francisco M. Borthagaray, Juan Pablo |
| author_facet | Acosta, Gabriel Belén, Eugenia Bersetche, Francisco M. Borthagaray, Juan Pablo |
| contents | We propose a deep learning approach to the obstacle problem inspired by the first-order system least-squares (FOSLS) framework. This method reformulates the problem as a convex minimization task; by simultaneously approximating the solution, gradient, and Lagrange multiplier, our approach provides a flexible, mesh-free alternative that scales efficiently to high-dimensional settings. Key theoretical contributions include the coercivity and local Lipschitz continuity of the proposed least-squares functional, along with convergence guarantees via $Γ$-convergence theory under mild regularity assumptions. Numerical experiments in dimensions up to 20 demonstrate the method's robustness and scalability, even on non-Lipschitz domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A deep first-order system least squares method for the obstacle problem Acosta, Gabriel Belén, Eugenia Bersetche, Francisco M. Borthagaray, Juan Pablo Numerical Analysis We propose a deep learning approach to the obstacle problem inspired by the first-order system least-squares (FOSLS) framework. This method reformulates the problem as a convex minimization task; by simultaneously approximating the solution, gradient, and Lagrange multiplier, our approach provides a flexible, mesh-free alternative that scales efficiently to high-dimensional settings. Key theoretical contributions include the coercivity and local Lipschitz continuity of the proposed least-squares functional, along with convergence guarantees via $Γ$-convergence theory under mild regularity assumptions. Numerical experiments in dimensions up to 20 demonstrate the method's robustness and scalability, even on non-Lipschitz domains. |
| title | A deep first-order system least squares method for the obstacle problem |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.19412 |