MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving
Abstract
Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC matches the accuracy of solutions using over more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs \textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.
Cite
@article{arxiv.2605.09288,
title = {MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving},
author = {Ethan Hsu and Hong Meng Yam and Ivan Ge},
journal= {arXiv preprint arXiv:2605.09288},
year = {2026}
}