FNWoS: Fractional Neural Walk-on-Spheres Methods for High-Dimensional PDEs Driven by $\alpha$-stable L\'{e}vy Process on Irregular Domains

Abstract

In this paper, we develop a highly parallel and derivative-free fractional neural walk-on-spheres method (FNWoS) for solving high-dimensional fractional Poisson equations on irregular domains. We first propose a simplified fractional walk-on-spheres (FWoS) scheme that replaces the high-dimensional normalized weight integral with a constant weight and adopts a correspondingly simpler sampling density, substantially reducing per-trajectory cost. To mitigate the slow convergence of standard Monte Carlo sampling, FNWoS is then proposed via integrating this simplified FWoS estimator, derived from the Feynman-Kac representation, with a neural network surrogate. By amortizing sampling effort over the entire domain during training, FNWoS achieves more accurate evaluation at arbitrary query points with dramatically fewer trajectories than classical FWoS. To further enhance efficiency in regimes where the fractional order $\alpha$ is close to 2 and trajectories become excessively long, we introduce a truncated path strategy with a prescribed maximum step count. Building on this, we propose a buffered supervision mechanism that caches training pairs and progressively refines their Monte Carlo targets during training, removing the need to precompute a highly accurate training set and yielding the buffered fractional neural walk-on-spheres method (BFNWoS). Extensive numerical experiments, including tests on irregular domains and problems with dimensions up to $1000$, demonstrate the accuracy, scalability, and computational efficiency of the proposed methods.