  Recently, artificial neural networks (ANNs) in conjunction with stochastic
gradient descent optimization methods have been employed to approximately
compute solutions of possibly rather high-dimensional partial differential
equations (PDEs). Very recently, there have also been a number of rigorous
mathematical results in the scientific literature which examine the
approximation capabilities of such deep learning based approximation algorithms
for PDEs. These mathematical results from the scientific literature prove in
part that algorithms based on ANNs are capable of overcoming the curse of
dimensionality in the numerical approximation of high-dimensional PDEs. In
these mathematical results from the scientific literature usually the error
between the solution of the PDE and the approximating ANN is measured in the
$L^p$-sense with respect to some $p \in [1,\infty)$ and some probability
measure. In many applications it is, however, also important to control the
error in a uniform $L^\infty$-sense. The key contribution of the main result of
this article is to develop the techniques to obtain error estimates between
solutions of PDEs and approximating ANNs in the uniform $L^\infty$-sense. In
particular, we prove that the number of parameters of an ANN to uniformly
approximate the classical solution of the heat equation in a region $ [a,b]^d $
for a fixed time point $ T \in (0,\infty) $ grows at most polynomially in the
dimension $ d \in \mathbb{N} $ and the reciprocal of the approximation
precision $ \varepsilon > 0 $. This shows that ANNs can overcome the curse of
dimensionality in the numerical approximation of the heat equation when the
error is measured in the uniform $L^\infty$-norm.
