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This paper introduces deep super ReLU networks (DSRNs) as a method forapproximating 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 deepneural networks (ReLU DNNs) cannot achieve this goal. DSRNs consist primarilyof ReLU DNNs, and several layers of the square of ReLU added at the end tosmooth the networks output. This approach retains the advantages of ReLU DNNs,leading to the straightforward training. The paper also proves the optimalityof DSRNs by estimating the VC-dimension of higher-order derivatives of DNNs,and obtains the generalization error in Sobolev spaces via an estimate of thepseudo-dimension of higher-order derivatives of DNNs.

Nearly Optimal Approximation Rates for Deep Super ReLU Networks on Sobolev Spaces