  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.
