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 are capable of representing Lagrange finite element functions of any order on simplicial meshes across arbitrary dimensions. We introduce a novel global formulation of the basis functions for Lagrange elements, grounded in a geometric decomposition of these elements and leveraging two essential properties of high-dimensional simplicial meshes and barycentric coordinate functions. 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.