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.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE); Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (math.NA) |
| Cite as: | arXiv:2605.09288 [cs.LG] |
| (or arXiv:2605.09288v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.09288 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Ethan Hsu [view email]
[v1]
Sun, 10 May 2026 03:32:46 UTC (5,635 KB)