  The development of new classification and regression algorithms based on
empirical risk minimization (ERM) over deep neural network hypothesis classes,
coined deep learning, revolutionized the area of artificial intelligence,
machine learning, and data analysis. In particular, these methods have been
applied to the numerical solution of high-dimensional partial differential
equations with great success. Recent simulations indicate that deep
learning-based algorithms are capable of overcoming the curse of dimensionality
for the numerical solution of Kolmogorov equations, which are widely used in
models from engineering, finance, and the natural sciences. The present paper
considers under which conditions ERM over a deep neural network hypothesis
class approximates the solution of a $d$-dimensional Kolmogorov equation with
affine drift and diffusion coefficients and typical initial values arising from
problems in computational finance up to error $\varepsilon$. We establish that,
with high probability over draws of training samples, such an approximation can
be achieved with both the size of the hypothesis class and the number of
training samples scaling only polynomially in $d$ and $\varepsilon^{-1}$. It
can be concluded that ERM over deep neural network hypothesis classes overcomes
the curse of dimensionality for the numerical solution of linear Kolmogorov
equations with affine coefficients.
