Operator Learning Using Weak Supervision from Walk-on-Spheres

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