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

Beyond black-boxes in Bayesian inverse problems and model validation: applications in solid mechanics of elastography