This article is concerned with the approximation and expressive powers of
deep neural networks. This is an active research area currently producing many
interesting papers. The results most commonly found in the literature prove
that neural networks approximate functions with classical smoothness to the
same accuracy as classical linear methods of approximation, e.g. approximation
by polynomials or by piecewise polynomials on prescribed partitions. However,
approximation by neural networks depending on n parameters is a form of
nonlinear approximation and as such should be compared with other nonlinear
methods such as variable knot splines or n-term approximation from
dictionaries. The performance of neural networks in targeted applications such
as machine learning indicate that they actually possess even greater
approximation power than these traditional methods of nonlinear approximation.
The main results of this article prove that this is indeed the case. This is
done by exhibiting large classes of functions which can be efficiently captured
by neural networks where classical nonlinear methods fall short of the task.
The present article purposefully limits itself to studying the approximation of
univariate functions by ReLU networks. Many generalizations to functions of
several variables and other activation functions can be envisioned. However,
even in this simplest of settings considered here, a theory that completely
quantifies the approximation power of neural networks is still lacking.

该论文研究了深度神经网络的近似和表达能力，证明了神经网络在目标应用中比传统的非线性近似方法具有更强的近似能力，其中逼近单变量函数的 ReLU 神经网络是研究的重点，然而，尚缺乏一种完全定量化神经网络近似能力的理论。