  In recent years, operator learning, particularly the DeepONet, has received
much attention for efficiently learning complex mappings between input and
output functions across diverse fields. However, in practical scenarios with
limited and noisy data, accessing the uncertainty in DeepONet predictions
becomes essential, especially in mission-critical or safety-critical
applications. Existing methods, either computationally intensive or yielding
unsatisfactory uncertainty quantification, leave room for developing efficient
and informative uncertainty quantification (UQ) techniques tailored for
DeepONets. In this work, we proposed a novel inference approach for efficient
UQ for operator learning by harnessing the power of the Ensemble Kalman
Inversion (EKI) approach. EKI, known for its derivative-free, noise-robust, and
highly parallelizable feature, has demonstrated its advantages for UQ for
physics-informed neural networks [28]. Our innovative application of EKI
enables us to efficiently train ensembles of DeepONets while obtaining
informative uncertainty estimates for the output of interest. We deploy a
mini-batch variant of EKI to accommodate larger datasets, mitigating the
computational demand due to large datasets during the training stage.
Furthermore, we introduce a heuristic method to estimate the artificial
dynamics covariance, thereby improving our uncertainty estimates. Finally, we
demonstrate the effectiveness and versatility of our proposed methodology
across various benchmark problems, showcasing its potential to address the
pressing challenges of uncertainty quantification in DeepONets, especially for
practical applications with limited and noisy data.
