  It is one of the most challenging issues in applied mathematics to
approximately solve high-dimensional partial differential equations (PDEs) and
most of the numerical approximation methods for PDEs in the scientific
literature suffer from the so-called curse of dimensionality in the sense that
the number of computational operations employed in the corresponding
approximation scheme to obtain an approximation precision $\varepsilon>0$ grows
exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$.
Recently, certain deep learning based approximation methods for PDEs have been
proposed and various numerical simulations for such methods suggest that deep
neural network (DNN) approximations might have the capacity to indeed overcome
the curse of dimensionality in the sense that the number of real parameters
used to describe the approximating DNNs grows at most polynomially in both the
PDE dimension $d\in\mathbb{N}$ and the reciprocal of the prescribed accuracy
$\varepsilon>0$. There are now also a few rigorous results in the scientific
literature which substantiate this conjecture by proving that DNNs overcome the
curse of dimensionality in approximating solutions of PDEs. Each of these
results establishes that DNNs overcome the curse of dimensionality in
approximating suitable PDE solutions at a fixed time point $T>0$ and on a
compact cube $[a,b]^d$ in space but none of these results provides an answer to
the question whether the entire PDE solution on $[0,T]\times [a,b]^d$ can be
approximated by DNNs without the curse of dimensionality. It is precisely the
subject of this article to overcome this issue. More specifically, the main
result of this work in particular proves for every $a\in\mathbb{R}$, $ b\in
(a,\infty)$ that solutions of certain Kolmogorov PDEs can be approximated by
DNNs on the space-time region $[0,T]\times [a,b]^d$ without the curse of
dimensionality.
