  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the
efficient solution of high-dimensional Poisson equations. Leveraging stochastic
representations and Walk-on-Spheres methods, we develop novel losses for neural
networks based on the recursive solution of Poisson equations on spheres inside
the domain. The resulting method is highly parallelizable and does not require
spatial gradients for the loss. We provide a comprehensive comparison against
competing methods based on PINNs, the Deep Ritz method, and (backward)
stochastic differential equations. In several challenging, high-dimensional
numerical examples, we demonstrate the superiority of NWoS in accuracy, speed,
and computational costs. Compared to commonly used PINNs, our approach can
reduce memory usage and errors by orders of magnitude. Furthermore, we apply
NWoS to problems in PDE-constrained optimization and molecular dynamics to show
its efficiency in practical applications.
