000165036 001__ 165036
000165036 005__ 20251204150239.0
000165036 0247_ $$2doi$$a10.1016/j.jcp.2025.114447
000165036 0248_ $$2sideral$$a146449
000165036 037__ $$aART-2025-146449
000165036 041__ $$aeng
000165036 100__ $$aAballay, Danilo
000165036 245__ $$aAn -adaptive finite element method using neural networks for parametric self-adjoint elliptic problems
000165036 260__ $$c2025
000165036 5060_ $$aAccess copy available to the general public$$fUnrestricted
000165036 5203_ $$aThis work proposes an -adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as those arising from self-adjoint elliptic problems. The objective of the NN optimization is to determine the mesh node locations. For simplicity, these locations are assumed to form a tensor product structure in higher dimensions. The method is designed to solve parametric partial differential equations (PDEs). The resulting parametric -adapted mesh generated by the NN is solved for each PDE parameter instance with a standard FEM. Consequently, the proposed approach retains the robustness and reliability guarantees of the FEM for each parameter instance, while the NN optimization adaptively adjusts the mesh node locations. The construction of FEM matrices and load vectors is implemented such that their derivatives with respect to mesh node locations, required for NN training, can be efficiently computed using automatic differentiation. However, the linear equation solver does not need to be differentiable, enabling the use of efficient, readily available ‘out-of-the-box’ solvers. The method’s performance is demonstrated on parametric one- and two-dimensional Poisson problems.
000165036 536__ $$9info:eu-repo/grantAgreement/ES/MICIU/CEX2021-001142-S$$9info:eu-repo/grantAgreement/ES/MICIU/PID2023-146668OA-100$$9info:eu-repo/grantAgreement/ES/MICIU/PID2023-146678OB-100
000165036 540__ $$9info:eu-repo/semantics/openAccess$$aby-nc$$uhttps://creativecommons.org/licenses/by-nc/4.0/deed.es
000165036 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion
000165036 700__ $$aFuentes, Federico
000165036 700__ $$aIligaray, Vicente
000165036 700__ $$0(orcid)0000-0002-3143-9097$$aOmella, Ángel J.$$uUniversidad de Zaragoza
000165036 700__ $$aPardo, David
000165036 700__ $$aSánchez, Manuel A.
000165036 700__ $$aTapia, Ignacio
000165036 700__ $$aUriarte, Carlos
000165036 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada
000165036 773__ $$g545 (2025), 114447 [22 pp.]$$pJ. comput. phys.$$tJournal of Computational Physics$$x0021-9991
000165036 8564_ $$s5302782$$uhttps://zaguan.unizar.es/record/165036/files/texto_completo.pdf$$yVersión publicada
000165036 8564_ $$s2100941$$uhttps://zaguan.unizar.es/record/165036/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada
000165036 909CO $$ooai:zaguan.unizar.es:165036$$particulos$$pdriver
000165036 951__ $$a2025-12-04-14:39:45
000165036 980__ $$aARTICLE