  This work presents novel extensions for combining two frameworks for
quantifying both aleatoric (i.e., irreducible) and epistemic (i.e., reducible)
sources of uncertainties in the modeling of engineered systems. The
data-consistent (DC) framework poses an inverse problem and solution for
quantifying aleatoric uncertainties in terms of pullback and push-forward
measures for a given Quantity of Interest (QoI) map. Unfortunately, a
pre-specified QoI map is not always available a priori to the collection of
data associated with system outputs. The data themselves are often polluted
with measurement errors (i.e., epistemic uncertainties), which complicates the
process of specifying a useful QoI. The Learning Uncertain Quantities (LUQ)
framework defines a formal three-step machine-learning enabled process for
transforming noisy datasets into samples of a learned QoI map to enable
DC-based inversion. We develop a robust filtering step in LUQ that can learn
the most useful quantitative information present in spatio-temporal datasets.
The learned QoI map transforms simulated and observed datasets into
distributions to perform DC-based inversion. We also develop a DC-based
inversion scheme that iterates over time as new spatial datasets are obtained
and utilizes quantitative diagnostics to identify both the quality and impact
of inversion at each iteration. Reproducing Kernel Hilbert Space theory is
leveraged to mathematically analyze the learned QoI map and develop a
quantitative sufficiency test for evaluating the filtered data. An illustrative
example is utilized throughout while the final two examples involve the
manufacturing of shells of revolution to demonstrate various aspects of the
presented frameworks.
