Active operator learning with predictive uncertainty quantification for partial differential equations

• URL: http://arxiv.org/abs/2503.03178v1
• Date: Wed, 05 Mar 2025 04:48:14 GMT
• Title: Active operator learning with predictive uncertainty quantification for partial differential equations
• Authors: Nick Winovich, Mitchell Daneker, Lu Lu, Guang Lin,
• Abstract summary: We develop a method for uncertainty quantification in deep operator networks (DeepONets) using predictive uncertainty estimates calibrated to model errors observed during training.<n>The uncertainty framework operates using a single network, in contrast to existing ensemble approaches, and introduces minimal overhead during training and inference.<n>We evaluate the uncertainty-equipped models on a series of partial differential equation (PDE) problems, and show that the model predictions are unbiased, non-skewed, and accurately reproduce solutions to the PDEs.
• Score: 6.519088943440059
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we develop a method for uncertainty quantification in deep operator networks (DeepONets) using predictive uncertainty estimates calibrated to model errors observed during training. The uncertainty framework operates using a single network, in contrast to existing ensemble approaches, and introduces minimal overhead during training and inference. We also introduce an optimized implementation for DeepONet inference (reducing evaluation times by a factor of five) to provide models well-suited for real-time applications. We evaluate the uncertainty-equipped models on a series of partial differential equation (PDE) problems, and show that the model predictions are unbiased, non-skewed, and accurately reproduce solutions to the PDEs. To assess how well the models generalize, we evaluate the network predictions and uncertainty estimates on in-distribution and out-of-distribution test datasets. We find the predictive uncertainties accurately reflect the observed model errors over a range of problems with varying complexity; simpler out-of-distribution examples are assigned low uncertainty estimates, consistent with the observed errors, while more complex out-of-distribution examples are properly assigned higher uncertainties. We also provide a statistical analysis of the predictive uncertainties and verify that these estimates are well-aligned with the observed error distributions at the tail-end of training. Finally, we demonstrate how predictive uncertainties can be used within an active learning framework to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures.

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