Solving Poisson Equations using Neural Walk-on-Spheres

TL;DR

Neural Walk-on-Spheres (NWoS) is a new neural PDE solver that efficiently solves high-dimensional Poisson equations using stochastic sphere-based methods, outperforming existing approaches in accuracy and computational cost.

Contribution

We introduce NWoS, a neural network method leveraging stochastic sphere techniques for high-dimensional Poisson equations, reducing memory and errors significantly.

Findings

NWoS achieves higher accuracy than PINNs and other methods.

NWoS reduces memory usage and errors by orders of magnitude.

NWoS demonstrates practical efficiency in PDE-constrained optimization and molecular dynamics.

Abstract

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.