  It is a challenging topic in applied mathematics to solve high-dimensional
nonlinear partial differential equations (PDEs). Standard approximation methods
for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense
that the number of computational operations of the approximation method grows
at least exponentially in the PDE dimension and with such methods it is
essentially impossible to approximately solve high-dimensional PDEs even when
the fastest currently available computers are used. However, in the last years
great progress has been made in this area of research through suitable deep
learning (DL) based methods for PDEs in which deep neural networks (DNNs) are
used to approximate solutions of PDEs. Despite the remarkable success of such
DL methods in simulations, it remains a fundamental open problem of research to
prove (or disprove) that such methods can overcome the COD in the approximation
of PDEs. However, there are nowadays several partial error analysis results for
DL methods for high-dimensional nonlinear PDEs in the literature which prove
that DNNs can overcome the COD in the sense that the number of parameters of
the approximating DNN grows at most polynomially in both the reciprocal of the
prescribed approximation accuracy $\varepsilon>0$ and the PDE dimension
$d\in\mathbb{N}$. In the main result of this article we prove that for all
$T,p\in(0,\infty)$ it holds that solutions
$u_d\colon[0,T]\times\mathbb{R}^d\to\mathbb{R}$, $d\in\mathbb{N}$, of
semilinear heat equations with Lipschitz continuous nonlinearities can be
approximated in the $L^p$-sense on space-time regions without the COD by DNNs
with the rectified linear unit (ReLU), the leaky ReLU, or the softplus
activation function. In previous articles similar results have been established
not for space-time regions but for the solutions $u_d(T,\cdot)$,
$d\in\mathbb{N}$, at the terminal time $T$.
