An -adaptive finite element method using neural networks for parametric self-adjoint elliptic problems

Aballay, Danilo ; Fuentes, Federico ; Iligaray, Vicente ; Omella, Ángel J. (Universidad de Zaragoza) ; Pardo, David ; Sánchez, Manuel A. ; Tapia, Ignacio ; Uriarte, Carlos

Resumen: This 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.

Idioma: Inglés

DOI: 10.1016/j.jcp.2025.114447

Año: 2025

Publicado en: Journal of Computational Physics 545 (2025), 114447 [22 pp.]
ISSN: 0021-9991

Financiación: info:eu-repo/grantAgreement/ES/MICIU/CEX2021-001142-S
Financiación: info:eu-repo/grantAgreement/ES/MICIU/PID2023-146668OA-100
Financiación: info:eu-repo/grantAgreement/ES/MICIU/PID2023-146678OB-100

Tipo y forma: Article (Published version)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)