  The present paper is motivated by one of the most fundamental challenges in
inverse problems, that of quantifying model discrepancies and errors. While
significant strides have been made in calibrating model parameters, the
overwhelming majority of pertinent methods is based on the assumption of a
perfect model. Motivated by problems in solid mechanics which, as all problems
in continuum thermodynamics, are described by conservation laws and
phenomenological constitutive closures, we argue that in order to quantify
model uncertainty in a physically meaningful manner, one should break open the
black-box forward model. In particular we propose formulating an undirected
probabilistic model that explicitly accounts for the governing equations and
their validity. This recasts the solution of both forward and inverse problems
as probabilistic inference tasks where the problem's state variables should not
only be compatible with the data but also with the governing equations as well.
Even though the probability densities involved do not contain any black-box
terms, they live in much higher-dimensional spaces. In combination with the
intractability of the normalization constant of the undirected model employed,
this poses significant challenges which we propose to address with a
linearly-scaling, double-layer of Stochastic Variational Inference. We
demonstrate the capabilities and efficacy of the proposed model in synthetic
forward and inverse problems (with and without model error) in elastography.
