Operator Learning Using Weak Supervision from Walk-on-Spheres

2026-03-01

AI Summary

This work proposes Walk-on-Spheres Neural Operator (WoS-NO), a novel framework that integrates the classical Walk-on-Spheres stochastic algorithm with neural operators to address key limitations of traditional neural PDE solvers. Conventional approaches often rely on expensive precomputed data or unstable physics-informed training, suffering from optimization challenges and high memory costs due to high-order derivatives. In contrast, WoS-NO enables a mesh-free, weakly supervised learning paradigm that requires neither high-order derivatives nor precomputed datasets. By leveraging Monte Carlo random walks to generate physical constraint signals, it learns solution operators for a class of PDEs and supports zero-shot generalization to unseen parameters and domains. Experiments demonstrate that, compared to standard physics-informed methods, WoS-NO reduces L₂ error by up to 8.75×, accelerates training by up to 6.31×, and cuts GPU memory usage by up to 2.97× under identical training budgets.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) that involves 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 results in a mesh-free framework that operates without expensive pre-computed datasets, avoids the need for 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

Problem

• neural PDE solvers
• physics-informed neural networks
• expensive data generation
• unstable optimization
• higher-order derivatives

Innovation

• Neural Operator
• Weak Supervision
• Walk-on-Spheres
• Physics-Informed Learning
• Mesh-Free PDE Solvers