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Artificial neural networks (ANNs) have become a very powerful tool in theapproximation of high-dimensional functions. Especially, deep ANNs, consistingof a large number of hidden layers, have been very successfully used in aseries of practical relevant computational problems involving high-dimensionalinput data ranging from classification tasks in supervised learning to optimaldecision problems in reinforcement learning. There are also a number ofmathematical results in the scientific literature which study the approximationcapacities of ANNs in the context of high-dimensional target functions. Inparticular, there are a series of mathematical results in the scientificliterature which show that sufficiently deep ANNs have the capacity to overcomethe curse of dimensionality in the approximation of certain target functionclasses in the sense that the number of parameters of the approximating ANNsgrows at most polynomially in the dimension $d \in \mathbb{N}$ of the targetfunctions under considerations. In the proofs of several of suchhigh-dimensional approximation results it is crucial that the involved ANNs aresufficiently deep and consist a sufficiently large number of hidden layerswhich grows in the dimension of the considered target functions. It is thetopic of this work to look a bit more detailed to the deepness of the involvedANNs in the approximation of high-dimensional target functions. In particular,the main result of this work proves that there exists a concretely specifiedsequence of functions which can be approximated without the curse ofdimensionality by sufficiently deep ANNs but which cannot be approximatedwithout the curse of dimensionality if the involved ANNs are shallow or notdeep enough.

Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality