Parameter calibration often assumes the computational model
can replicate the true physical mechanism behind data generation. In
practice, however, computational models rely on parameterizations,
assumptions, and constitutive relations that entail significant model
structural error. Ignoring such model errors can lead to overconfident
calibrations and poor predictive capability, even when high-quality
data are used for calibration. It is thus crucial to quantify and
propagate uncertainty due to model error, and to differentiate it from
parametric uncertainty and data noise. Traditional approaches
accommodate model error through discrepancy terms that are only
available for model output quantities used for calibration, and
generally do not preserve physical constraints in subsequent
predictions. The ability to extrapolate to other predictive quantities
and to retain certain physical properties (e.g. conservation
principles, positivity constraints) is often required in physical
science and engineering applications.
We develop a stochastically embedded model correction approach
that enables these qualities, and illustrate computational methods for
Bayesian inference of the correction terms together with model
parameters [Sargsyan et al., 2018]. Representing the correction terms
using polynomial chaos expansions, the new formulation becomes a
density estimation problem [Sargsyan et al., 2015], and allows efficient
quantification, propagation, and decomposition of uncertainty that
includes contributions from data noise, parameter posterior
uncertainty, and model error. The framework provides principled tools
for the analyst, e.g. to examine the utility of corrections to
specific suspect model components, and to identify the model
components where model improvements are relevant for agreement with
the data.
We demonstrate the key strengths of this method on realistic
engineering applications, including climate models, chemical
kinetics, and reacting flow simulations from a supersonic jet engine design [Huan et al. 2017].