Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called \emph{Walk-on-Spheres Neural Operator (WoS-NO)} which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75$\times$ improvement in $L_2$-error compared to standard physics-informed training schemes, up to 6.31$\times$ improvement in training speed, and reductions of up to 2.97$\times$ in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

翻译：训练神经偏微分方程求解器通常受限于昂贵的数据生成过程，或不稳定的物理信息神经网络（PINN）——后者因涉及高阶导数而面临具有挑战性的优化地形。为解决这一问题，我们提出一种替代方法，利用蒙特卡洛方法将偏微分方程的解估计为一个随机过程，从而在训练过程中提供弱监督。借助球面行走方法，我们引入了一种名为“球面行走神经算子”的学习方案，该方案利用来自球面行走的弱监督来训练任意给定的神经算子。我们提出，通过利用球面行走算法中的随机表示，在偏微分方程实例的分布上分摊蒙特卡洛行走的计算成本，从而在训练过程中生成廉价、含噪声的偏微分方程解估计。这被形式化为一个无数据的物理信息目标函数，其中神经算子被训练以回归这些弱监督信号，从而使算子能够学习整个偏微分方程族的广义解映射。该策略无需昂贵的预计算数据集，避免了计算内存密集且不稳定的损失函数中的高阶导数，并展示了对新偏微分方程参数和领域的零样本泛化能力。实验表明，在相同训练步数下，与标准的物理信息训练方案相比，我们的方法在$L_2$误差上实现了高达8.75倍的提升，训练速度提高了高达6.31倍，GPU内存消耗降低了高达2.97倍。代码发布于 https://github.com/neuraloperator/WoS-NO。