  This paper introduces deep super ReLU networks (DSRNs) as a method for
approximating functions in Sobolev spaces measured by Sobolev norms $W^{m,p}$
for $m\in\mathbb{N}$ with $m\ge 2$ and $1\le p\le +\infty$. Standard ReLU deep
neural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarily
of ReLU DNNs, and several layers of the square of ReLU added at the end to
smooth the networks output. This approach retains the advantages of ReLU DNNs,
leading to the straightforward training. The paper also proves the optimality
of DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs,
and obtains the generalization error in Sobolev spaces via an estimate of the
pseudo-dimension of higher-order derivatives of DNNs.
