  Artificial neural networks (ANNs) have very successfully been used in
numerical simulations for a series of computational problems ranging from image
classification/image recognition, speech recognition, time series analysis,
game intelligence, and computational advertising to numerical approximations of
partial differential equations (PDEs). Such numerical simulations suggest that
ANNs have the capacity to very efficiently approximate high-dimensional
functions and, especially, indicate that ANNs seem to admit the fundamental
power to overcome the curse of dimensionality when approximating the
high-dimensional functions appearing in the above named computational problems.
There are a series of rigorous mathematical approximation results for ANNs in
the scientific literature. Some of them prove convergence without convergence
rates and some even rigorously establish convergence rates but there are only a
few special cases where mathematical results can rigorously explain the
empirical success of ANNs when approximating high-dimensional functions. The
key contribution of this article is to disclose that ANNs can efficiently
approximate high-dimensional functions in the case of numerical approximations
of Black-Scholes PDEs. More precisely, this work reveals that the number of
required parameters of an ANN to approximate the solution of the Black-Scholes
PDE grows at most polynomially in both the reciprocal of the prescribed
approximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in
\mathbb{N}$. We thereby prove, for the first time, that ANNs do indeed overcome
the curse of dimensionality in the numerical approximation of Black-Scholes
PDEs.
