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.

翻译：本文提出深度超ReLU网络（DSRNs）作为一种在Sobolev空间中逼近函数的方法，该空间由Sobolev范数 $W^{m,p}$ 衡量，其中 $m\in\mathbb{N}$ 且 $m\ge 2$，$1\le p\le +\infty$。标准ReLU深度神经网络（ReLU DNNs）无法实现这一目标。DSRNs主要由ReLU DNNs构成，并在末端添加若干层ReLU的平方以平滑网络输出。该方法保留了ReLU DNNs的优势，从而便于训练。本文还通过估计DNNs高阶导数的VC维证明了DSRNs的最优性，并通过估计DNNs高阶导数的伪维得到了Sobolev空间中的泛化误差。