Computational models usually carry misspecification due to different parameterizations
and assumptions. Ignoring such model errors can lead to poor predictive capability,
even when high-quality data are used. 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 calibration quantities,
and generally do not preserve physical model constraints in subsequent
predictions. The ability to extrapolate to other predictive quantities
and to retain certain physical properties are often desirable and even required in many physical science and
engineering applications. We develop a stochastically
embedded model correction approach that enables these qualities,
and perform Bayesian inference of the correction
terms together with model parameters. Employing polynomial chaos expansions
to represent the correction terms, the approach allows
efficient quantification, propagation, and decomposition of
uncertainty that includes contributions from data noise,
parameter posterior uncertainty, and model error.