基于球面游走弱监督的算子学习方法（注：Walk-on-Spheres是计算数学中用于求解偏微分方程的蒙特卡洛方法，在此语境下保留专业术语特性，采用"球面游走"这一学界通用译法，既准确传达随机游走于球面的核心意象，又符合计算数学领域的中文表述规范。）

摘要

训练神经偏微分方程求解器常受限于昂贵的数据生成成本，或面临因高阶导数导致优化空间复杂而难以稳定的物理信息神经网络（PINN）。为解决该问题，我们提出一种基于蒙特卡洛方法的替代方案，通过将偏微分方程解估计为随机过程，为训练过程提供弱监督。借助球面行走法，我们提出名为球面行走神经算子（WoS-NO）的学习框架，利用WoS生成的弱监督信号训练任意给定的神经算子。通过WoS算法的随机表示，我们将蒙特卡洛行走的计算成本分摊到偏微分方程实例的分布上，在训练过程中生成廉价且含噪声的偏微分方程解估计值。该方法被形式化为无数据的物理信息优化目标，通过训练神经算子回归这些弱监督信号，使其能够学习整个偏微分方程族系的广义解映射。该策略无需昂贵的预计算数据集，避免了内存密集且不稳定的高阶导数损失函数计算，并展现出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，我们的方法相比标准物理信息训练方案L2误差最高提升8.75倍，训练速度最高提升6.31倍，GPU内存消耗最高降低2.97倍。代码发布于https://github.com/neuraloperator/WoS-NO。

English

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 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 8.75times improvement in L_2-error compared to standard physics-informed training schemes, up to 6.31times improvement in training speed, and reductions of up to 2.97times in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

Operator Learning Using Weak Supervision from Walk-on-Spheres

摘要

Support