Beyond Black-Boxes in Bayesian Inverse Problems and Model Validation: Applications in Solid Mechanics of Elastography

Abstract

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 linearlyscaling, 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.

CUQIpy: II. Computational uncertainty quantification for PDE-based inverse problems in Python

“…Utilizing the flexibility of CUQIpy modeling framework, we demonstrate combining multiple datasets, namely, data resulting from multiple injection patterns in the EIT example. For simplicity, we assume these models are exact and we leave treatment of forward model error [7,10] for future investigation. We emphasize that the CUQIpy framework is general to explore other types of unknown parameterizations, priors (e.g.…”

A Perspective on Digital Knowledge Representation in Materials Science and Engineering

“…This is particularly true for situations, where a significant model bias is expected (as is often the case for failure models due to an insufficient model in these extreme scenarios or due to randomness in crack localization). Applying methods such as FEMU‐F [ 114 ] with stochastic models [ 115 ] allows improving the quality of the calibration. The essential idea is to interpret the finite element model (FEM) solution as a stochastic variable (or random field when taking into account correlations) that is identified.…”

Bayesian inference for random field parameters with a goal-oriented quality control of the PGD forward model’s accuracy

“…2. In the works of [14] and [15], it was also used to infer material properties. Among the limitations of the method is the fact that, unlike MCMC, the inference results can be biased and their accuracy strongly depends on the family of approximating distributions employed.…”