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.

提出了一种名为Neural Walk-on-Spheres（NWoS）的新型神经PDE求解器，用于高维Poisson方程的高效求解。通过利用随机表示和Walk-on-Spheres方法，基于球内泊松方程的递归求解，我们开发了用于神经网络的新型损失函数。该方法具有高度的可并行性，且不需要损失函数的空间梯度。通过与基于PINNs，Deep Ritz方法和（反向）随机微分方程的竞争方法进行全面比较，在多个具有挑战性的高维数值示例中，我们证明了NWoS在准确性、速度和计算成本方面的优越性。与常用的PINNs相比，我们的方法可以将内存使用和误差降低数个数量级。此外，我们还将NWoS应用于PDE约束优化和分子动力学问题中，展示了其在实际应用中的高效性。

使用神经行星行走方法解决泊松方程