Deep neural networks and finite elements of any order on arbitrary dimensions

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.