Deep Ritz- Finite element methods: Neural network methods trained with finite elements

被引：0

作者：

Grekas, Georgios ^{[1
,2
]}

Makridakis, Charalambos G. ^{[1
,3
,4
]}

机构：

[1] FORTH, Inst Appl & Computat Math, Iraklion 70013, Crete, Greece

[2] King Abdullah Univ Sci & Technol KAUST, Comp Elect Math Sci & Engn Div, Thuwal 239556900, Saudi Arabia

[3] Univ Crete, DMAM, Iraklion, Greece

[4] Univ Sussex, MPS, Brighton BN1 9QH, England

来源：

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING | 2025年 / 437卷

关键词：

Neural networks; Finite elements; PINNS; CONVERGENCE; APPROXIMATION; ALGORITHM;

D O I：

10.1016/j.cma.2025.117798

中图分类号：

T [工业技术];

学科分类号：

08 ;

摘要：

While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains Omega subset of Rd, d = 1, 2, 3 in association with more standard finite elements. We suggest to connect finite elements and neural network approximations through training, i.e., using finite element spaces to compute the integrals appearing in the loss functionals. This approach, retains the simplicity of classical neural network methods for PDEs, uses well established finite element tools (and software) to compute the integrals involved and it gains in efficiency and accuracy. We demonstrate that the proposed methods are stable and furthermore, we establish that the resulting approximations converge to the solutions of the PDE. Numerical results indicating the efficiency and robustness of the proposed algorithms are presented.

引用

页数：15

共 48 条

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