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.

翻译：偏微分方程求解器是科学计算的基石，但其实际部署受限于计算资源。经典的蒙特卡罗求解器（如球面游走法）无偏且与几何无关，但速度缓慢；而基于学习的求解器虽快，却存在偏差且对分布偏移敏感。我们提出**MC$^2$**，一种混合型球面游走-神经网络（WoS-NN）偏微分方程求解器，将低预算蒙特卡罗解视为真实场的一种结构化估计量，并通过学习单次前向传播的神经校正来恢复高保真解。MC$^2$的精度可媲美使用超过**1000倍**蒙特卡罗计算量的解，优于所有被评估的经典解法、去噪方法及神经算子基线。为实现有限计算资源下偏微分方程求解的可重复研究，我们同时发布**PDEZoo**——迄今最大规模的标准化椭圆型偏微分方程基准测试集：包含200万个跨越五种椭圆族及无限几何组合的偏微分方程，并提供解析真解与多预算蒙特卡罗轨迹。**MC$^2$**与**PDEZoo**共同：（1）从实验上证实有限样本蒙特卡罗误差具有结构化特性，可通过单次前向传播学习并校正；（2）表明我们求解偏微分方程的速度可比纯球面游走法快约**1000倍**；（3）为该领域提供了此前所缺乏的评估基础设施。