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 tasks 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\%.

运算符学习在科学和工程应用领域中得到了越来越广泛的应用，其中许多需要经过校准的不确定性量化。本文提出了一种风险控制分位数神经运算符，即无分布、有限样本的函数校准型预测方法，通过实验证实了我们的理论结果，显示了校准的覆盖率和高效的不确定性范围，优于基准方法。