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We propose two bounded comparison metrics that may be implemented toarbitrary dimensions in regression tasks. One quantifies the structure ofuncertainty and the other quantifies the distribution of uncertainty. Thestructure metric assesses the similarity in shape and location of uncertaintywith the true error, while the distribution metric quantifies the supportedmagnitudes between the two. We apply these metrics to Gaussian Processes (GPs),Ensemble Deep Neural Nets (DNNs), and Ensemble Deep Neural Operators (DNOs) onhigh-dimensional and nonlinear test cases. We find that comparing a model'suncertainty estimates with the model's squared error provides a compellingground truth assessment. We also observe that both DNNs and DNOs, especiallywhen compared to GPs, provide encouraging metric values in high dimensions witheither sparse or plentiful data.

Structure and Distribution Metric for Quantifying the Quality of  Uncertainty: Assessing Gaussian Processes, Deep Neural Nets, and Deep Neural  Operators for Regression