MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving

TL;DR

MC$^2$ introduces a hybrid Monte Carlo-neural PDE solver that significantly improves accuracy and speed over classical methods, and provides a large standardized PDE benchmark for reproducible research.

Contribution

The paper presents MC$^2$, a novel neural correction method for Monte Carlo PDE solvers, and releases PDEZoo, the largest elliptic PDE benchmark dataset.

Findings

MC$^2$ matches high-fidelity solutions with 1000x less Monte Carlo compute.

Finite-sample Monte Carlo error is learnable and correctable in a single pass.

PDEZoo enables reproducible evaluation of PDE solvers across diverse geometries.

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$^2$}, 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$^2$ matches the accuracy of solutions using over $1000\times$ 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$^2$} 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 $\sim$\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.