ReLU Deep Neural Networks and Linear Finite Elements

In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the simplicial linear finite element method (FEM). We first consider the special case of FEM. By exploring the DNN representation of its nodal basis functions, we present a ReLU DNN representation of CPWL in FEM. We theoretically establish that at least $2$ hidden layers are needed in a ReLU DNN to represent any linear finite element functions in $\Omega \subseteq \mathbb{R}^d$ when $d\ge2$. Consequently, for $d=2,3$ which are often encountered in scientific and engineering computing, the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN. Then we include a detailed account on how a general CPWL in $\mathbb R^d$ can be represented by a ReLU DNN with at most $\lceil\log_2(d+1)\rceil$ hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a representation. Furthermore, using the relationship between DNN and FEM, we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications. Finally, as a proof of concept, we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.