MC$^2$：用于快速求解椭圆 PDE 的蒙特卡洛校正方法
MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving
AI 导读
MC$^2$ 是一种混合神经网络-蒙特卡洛 PDE 求解器，将低预算蒙特卡洛解视为结构化估计量，通过单次前向神经校正恢复高保真解，精度相当于超过 1000 倍蒙特卡洛样本的解。相比 WoS 在匹配精度下提速约 218 倍，推理时间较 SOTA 扩散方法减少约 171 倍且精度更高。团队同时发布含 200 万个 PDE、覆盖 5 个椭圆族的基准 PDEZoo，代码与数据已开源。
正文 · 原文
Abstract:Partial differential equation (PDE) solvers underpin modern computer graphics, geometry processing, and scientific computing. Classical Monte Carlo PDE solvers such as Walk-on-Spheres (WoS) provide mesh-free estimations that are unbiased, but with accuracy limited by sampling and numerical approximation. Learned solvers are fast but biased and brittle under distribution shift. We therefore present MC$^2$, a hybrid Neural Network-Monte Carlo 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 samples, outperforming all evaluated mesh-free classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release PDEZoo, the largest standardized elliptic PDE benchmark to date: 2 million PDEs spanning 5 elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together, MC$^2$ and 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$218$\times$ faster than WoS at matched accuracy and achieve a $\sim$171$\times$ reduction in inference time relative to state-of-the-art diffusion methods while attaining higher accuracy, and (3) provide the reproducible evaluation infrastructure the field has so far lacked.
Code & Data: this https URL
| Comments: | Accepted to NeurIPS 2026 |
| 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:2605.09288v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.09288 arXiv-issued DOI via DataCite |
Submission history
From: Ethan Hsu [view email]
[v1]
Sun, 10 May 2026 03:32:46 UTC (5,635 KB)
[v2]
Tue, 10 Nov 2026 02:30:51 UTC (7,172 KB)
来源：arXiv cs.AI · arxiv.org