Gradient-Free Physics-informed Operator Learning using Walk-on-Spheres

We introduce a Monte Carlo-based neural operator learning framework, WoS-GINO, for solving families of elliptic partial differential equations (PDEs) on irregular geometries. Our gradient-free physics-informed loss is based on noisy but cheap Walk-on-Spheres (WoS) simulations with a small number of trajectories. Due to the unbiasedness of the estimates, the resulting regression objective still recovers the ground-truth operator, while eliminating the need for data generation via expensive solvers and circumventing the optimization challenges of existing physics-informed losses. In particular, our approach is mesh-free and amortizes the cost of WoS across PDEs, allowing for zero-shot generalization to unseen PDE parameters, geometries, and discretizations during inference. In our experiments, we show that the resulting method achieves 10$\times$ improvement over naive WoS solvers and is at least 23$\times$ faster.