Mathematical Models and Methods in Applied Sciences· 2026Q1
Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions
- 2citations
- Q1SCImago
- 2026year
Short summary
Deep neural networks (DNNs) with ReLU or ReLU 2 activations can precisely represent Lagrange finite element functions of any order on arbitrary simplicial meshes in any dimension.
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Key points
- Deep neural networks (DNNs) with ReLU/ReLU 2 activations can represent Lagrange finite element functions of any order.
- This representation holds for arbitrary simplicial meshes in any dimension.
- Novel formulations for global basis function expression are introduced, based on geometric decomposition.
- The findings show DNNs can systematically generate continuous piecewise polynomial functions.
AI-generated from the title and abstract; the full text is not read.
Abstract
In this study, we establish that deep neural networks employing ReLU and ReLU 2 activation functions can rigorously represent Lagrange finite element functions of any order on arbitrary simplicial meshes in any dimension. We introduce two novel formulations for globally expressing the basis functions of Lagrange elements, tailored for both specific and arbitrary meshes. These formulations are based on a geometric decomposition of the elements, incorporating several insightful and essential properties of high-dimensional simplicial meshes, barycentric coordinate functions, and global basis functions of linear elements. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions on both specific and arbitrary simplicial meshes.
The authors' abstract, as published at the source. Mathematical Models and Methods in Applied Sciences, 2026 · DOI ↗
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Field: Computational Mechanics
Computational MechanicsEngineering