  Over the last few years deep artificial neural networks (DNNs) have very
successfully been used in numerical simulations for a wide variety of
computational problems including computer vision, image classification, speech
recognition, natural language processing, as well as computational
advertisement. In addition, it has recently been proposed to approximate
solutions of partial differential equations (PDEs) by means of stochastic
learning problems involving DNNs. There are now also a few rigorous
mathematical results in the scientific literature which provide error estimates
for such deep learning based approximation methods for PDEs. All of these
articles provide spatial error estimates for neural network approximations for
PDEs but do not provide error estimates for the entire space-time error for the
considered neural network approximations. It is the subject of the main result
of this article to provide space-time error estimates for DNN approximations of
Euler approximations of certain perturbed differential equations. Our proof of
this result is based (i) on a certain artificial neural network (ANN) calculus
and (ii) on ANN approximation results for products of the form $[0,T]\times
\mathbb{R}^d\ni (t,x)\mapsto tx\in \mathbb{R}^d$ where $T\in (0,\infty)$, $d\in
\mathbb{N}$, which we both develop within this article.
