Solving Poisson Equations using Neural Walk-on-Spheres
This addresses the problem of efficiently solving high-dimensional Poisson equations for researchers in computational science, with incremental improvements in method integration.
The paper tackles solving high-dimensional Poisson equations by proposing Neural Walk-on-Spheres (NWoS), a neural PDE solver that reduces memory usage and errors by orders of magnitude compared to PINNs, while improving accuracy, speed, and computational costs.
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.