Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions

TL;DR

This paper demonstrates that deep neural networks with ReLU and ReLU^2 activations can systematically represent high-order finite element functions on arbitrary-dimensional meshes, bridging neural networks and finite element methods.

Contribution

It introduces novel formulations for representing Lagrange finite element basis functions using deep neural networks in arbitrary dimensions and mesh types.

Findings

Neural networks can represent finite element basis functions of any order.

New formulations enable systematic generation of continuous piecewise polynomial functions.

First demonstration of neural networks approximating high-order finite element spaces on arbitrary meshes.

Abstract

In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. 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 or arbitrary simplicial meshes.