  We present a deep learning algorithm for the numerical solution of parametric
families of high-dimensional linear Kolmogorov partial differential equations
(PDEs). Our method is based on reformulating the numerical approximation of a
whole family of Kolmogorov PDEs as a single statistical learning problem using
the Feynman-Kac formula. Successful numerical experiments are presented, which
empirically confirm the functionality and efficiency of our proposed algorithm
in the case of heat equations and Black-Scholes option pricing models
parametrized by affine-linear coefficient functions. We show that a single deep
neural network trained on simulated data is capable of learning the solution
functions of an entire family of PDEs on a full space-time region. Most
notably, our numerical observations and theoretical results also demonstrate
that the proposed method does not suffer from the curse of dimensionality,
distinguishing it from almost all standard numerical methods for PDEs.
