  We propose two bounded comparison metrics that may be implemented to
arbitrary dimensions in regression tasks. One quantifies the structure of
uncertainty and the other quantifies the distribution of uncertainty. The
structure metric assesses the similarity in shape and location of uncertainty
with the true error, while the distribution metric quantifies the supported
magnitudes between the two. We apply these metrics to Gaussian Processes (GPs),
Ensemble Deep Neural Nets (DNNs), and Ensemble Deep Neural Operators (DNOs) on
high-dimensional and nonlinear test cases. We find that comparing a model's
uncertainty estimates with the model's squared error provides a compelling
ground truth assessment. We also observe that both DNNs and DNOs, especially
when compared to GPs, provide encouraging metric values in high dimensions with
either sparse or plentiful data.
