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 (CoD) 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 CoD in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension $d \in \N$ and the reciprocal of the prescribed approximation accuracy $\varepsilon >0$. There are now also a few rigorous mathematical results in the scientific literature which substantiate this conjecture by proving that DNNs overcome the CoD in approximating solutions of PDEs. Each of these results establishes that DNNs overcome the CoD in approximating suitable PDE solutions at a fixed time point $T >0$ and on a compact cube $[a, b]^d$ 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 CoD.
In this talk we show that for every $a \in \R$, $ b \in (a, \infty)$ solutions of suitable Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0, T] \times [a, b]^d$ without the CoD.