arxiv:2603.01193

Operator Learning Using Weak Supervision from Walk-on-Spheres

Published on Mar 1

· Submitted by

Hrishikesh Viswanath on Mar 9

Purdue University

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Abstract

Neural PDE solvers trained with Monte Carlo-based weak supervision achieve improved accuracy, faster training, and reduced memory usage compared to traditional physics-informed methods.

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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

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hrishivish23

Paper submitter about 10 hours ago

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