# Operator Learning Using Weak Supervision from Walk-on-Spheres

arXiv: https://arxiv.org/abs/2603.01193 · HTML: https://mlsanity.com/paper/2603.01193/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 49 references that are in the map.
- First posted on arXiv 1 Mar 2026.

## Builds on (most cited first, 23 of 23)

- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh](https://mlsanity.com/paper/2211.01604/) (2022)
- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients](https://mlsanity.com/paper/2201.13240/) (2022)
- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions](https://mlsanity.com/paper/2302.11815/) (2023)
- [Boundary Value Caching for Walk on Spheres](https://mlsanity.com/paper/2302.11825/) (2023)
- [Robust Biharmonic Skinning Using Geometric Fields](https://mlsanity.com/paper/2406.00238/) (2024)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)
- [Multi-Level Monte Carlo Training of Neural Operators](https://mlsanity.com/paper/2505.12940/) (2025)
- [DeepRWCap: Neural-Guided Random-Walk Capacitance Solver for IC Design](https://mlsanity.com/paper/2511.06831/) (2025)
- [Solving Inverse PDE Problems using Grid-Free Monte Carlo Estimators](https://mlsanity.com/paper/2208.02114/) (2022)
- [Solving partial differential equations in participating media](https://mlsanity.com/paper/2506.08237/) (2025)

## Built on by (most cited first, 2 of 2)

- [Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds](https://mlsanity.com/paper/2606.00643/) (2026)
- [PTNO: Training Neural Operators with Noisy Monte Carlo Estimates for Particle Transport Problems](https://mlsanity.com/paper/2609.40090/) (2026)

## Abstract (arXiv)

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 8.75× improvement in L2-error compared to standard physics-informed training schemes, up to 6.31× improvement in training speed, and reductions of up to 2.97× in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO