Hierarchical Bayesian learning framework for multi-level modeling using multi-level data

A hierarchical Bayesian learning framework is proposed to account for multi-level modeling in structural dynamics. In multi-level modeling the system is considered as a hierarchy of lower level models, starting at the lowest material level, progressing to the component level, then the subsystem level, before ending up to the system level. Bayesian modeling and uncertainty quantification techniques based on measurements that rely on data collected only at the system level cover a quite limited number of component/subsystem operating conditions that are far from representing the full spectrum of system operating conditions. In addition, the large number of models and parameters involved from the lower to higher modeling levels of the system, constitutes this approach inappropriate for simultaneously and reliably quantifying the uncertainties at the different modeling levels. In this work, comprehensive hierarchical Bayesian learning tools are proposed to account for uncertainties through the multi-level modeling process. The uncertainty is embedded within the structural model parameters by introducing a probability model for these parameters that depend on hyperparameters. An important issue that has to be accounted for is that parameters of models at lower levels are shared at the subsystem and system levels. This necessitates a parameter inference process that takes into account data from different modeling levels. Accurate and insightful asymptotic approximations are developed, substantially reducing the computational effort required in the parameter uncertainty quantification procedure. The uncertainties inferred based on datasets available from the different levels of model hierarchy are propagated through the different levels of the system to predict uncertainties and confidence levels of output quantities of interest. A simple dynamical system consisting of components and subsystems is employed to demonstrate the effectiveness of the proposed method.