  Operator learning has been increasingly adopted in scientific and engineering
applications, many of which require calibrated uncertainty quantification.
Since the output of operator learning is a continuous function, quantifying
uncertainty simultaneously at all points in the domain is challenging. Current
methods consider calibration at a single point or over one scalar function or
make strong assumptions such as Gaussianity. We propose a risk-controlling
quantile neural operator, a distribution-free, finite-sample functional
calibration conformal prediction method. We provide a theoretical calibration
guarantee on the coverage rate, defined as the expected percentage of points on
the function domain whose true value lies within the predicted uncertainty
ball. Empirical results on a 2D Darcy flow and a 3D car surface pressure
prediction task validate our theoretical results, demonstrating calibrated
coverage and efficient uncertainty bands outperforming baseline methods. In
particular, on the 3D problem, our method is the only one that meets the target
calibration percentage (percentage of test samples for which the uncertainty
estimates are calibrated) of 98%.
