  Numerical experiments indicate that deep learning algorithms overcome the
curse of dimensionality when approximating solutions of semilinear PDEs. For
certain linear PDEs and semilinear PDEs with gradient-independent
nonlinearities this has also been proved mathematically, i.e., it has been
shown that the number of parameters of the approximating DNN increases at most
polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of
the prescribed accuracy $\epsilon\in (0,1)$. The main contribution of this
paper is to rigorously prove for the first time that deep neural networks can
also overcome the curse dimensionality in the approximation of a certain class
of nonlinear PDEs with gradient-dependent nonlinearities.
