Simultaneous Neural Network Approximation for Smooth Functions

We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our approximation results are nonasymptotic in the sense that the error bounds are explicitly characterized in terms of both the width and depth of the networks simultaneously with all involved constants explicitly determined.} Namely, for $f\in C^s([0,1]^d)$, we show that deep ReLU networks of width $\mathcal{O}(N\log{N})$ and of depth $\mathcal{O}(L\log{L})$ can achieve a nonasymptotic approximation rate of $\mathcal{O}(N^{-2(s-1)/d}L^{-2(s-1)/d})$ with respect to the $\mathcal{W}^{1,p}([0,1]^d)$ norm for $p\in[1,\infty)$. If either the ReLU function or its square is applied as activation functions to construct deep neural networks of width $\mathcal{O}(N\log{N})$ and of depth $\mathcal{O}(L\log{L})$ to approximate $f\in C^s([0,1]^d)$, the approximation rate is $\mathcal{O}(N^{-2(s-n)/d}L^{-2(s-n)/d})$ with respect to the $\mathcal{W}^{n,p}([0,1]^d)$ norm for $p\in[1,\infty)$. An extension of similar approximation results is also provided for target functions in the H\"{o}lder space.