  The combination of Monte Carlo methods and deep learning has recently led to
efficient algorithms for solving partial differential equations (PDEs) in high
dimensions. Related learning problems are often stated as variational
formulations based on associated stochastic differential equations (SDEs),
which allow the minimization of corresponding losses using gradient-based
optimization methods. In respective numerical implementations it is therefore
crucial to rely on adequate gradient estimators that exhibit low variance in
order to reach convergence accurately and swiftly. In this article, we
rigorously investigate corresponding numerical aspects that appear in the
context of linear Kolmogorov PDEs. In particular, we systematically compare
existing deep learning approaches and provide theoretical explanations for
their performances. Subsequently, we suggest novel methods that can be shown to
be more robust both theoretically and numerically, leading to substantial
performance improvements.
