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In this article we study high-dimensional approximation capacities of shallowand deep artificial neural networks (ANNs) with the rectified linear unit(ReLU) activation. In particular, it is a key contribution of this work toreveal that for all $a,b\in\mathbb{R}$ with $b-a\geq 7$ we have that thefunctions $[a,b]^d\ni x=(x_1,\dots,x_d)\mapsto\prod_{i=1}^d x_i\in\mathbb{R}$for $d\in\mathbb{N}$ as well as the functions $[a,b]^d\ni x =(x_1,\dots,x_d)\mapsto\sin(\prod_{i=1}^d x_i) \in \mathbb{R} $ for $ d \in \mathbb{N} $can neither be approximated without the curse of dimensionality by means ofshallow ANNs nor insufficiently deep ANNs with ReLU activation but can beapproximated without the curse of dimensionality by sufficiently deep ANNs withReLU activation. We show that the product functions and the sine of the productfunctions are polynomially tractable approximation problems among theapproximating class of deep ReLU ANNs with the number of hidden layers beingallowed to grow in the dimension $ d \in \mathbb{N} $. We establish the aboveoutlined statements not only for the product functions and the sine of theproduct functions but also for other classes of target functions, inparticular, for classes of uniformly globally bounded $ C^{ \infty }$-functions with compact support on any $[a,b]^d$ with $a\in\mathbb{R}$,$b\in(a,\infty)$. Roughly speaking, in this work we lay open that simpleapproximation problems such as approximating the sine or cosine of productscannot be solved in standard implementation frameworks by shallow orinsufficiently deep ANNs with ReLU activation in polynomial time, but can beapproximated by sufficiently deep ReLU ANNs with the number of parametersgrowing at most polynomially.

The necessity of depth for artificial neural networks to approximate  certain classes of smooth and bounded functions without the curse of  dimensionality