Combining data and physical models for probabilistic analysis: A Bayesian Augmented Space Learning perspective

The traditional methods for probabilistic analysis of physical systems often follow a non-intrusive scheme with, random samples for stochastic model parameters generated in the outer loop, and for each sample, physical model (described by PDEs) solved in the inner loop using, e.g., finite element method (FEM). Two of the biggest challenges when applying probabilistic methods are the high computational burden due to the repeated calls of the expensive-to-estimate computational models, and the difficulties of integrating the numerical errors from both loops. To overcome these challenges, we present a new framework for transforming the PDEs with stochastic parameters into equivalent deterministic PDEs, and then devise a statistical inference method, called Bayesian Augmented Space Learning (BASL), for inferring the probabilistic descriptors of the model responses with the combination of measurement data and physical models. With the two sources of information available, only a one-step Bayesian inference needs to be performed, and the numerical errors are summarized by posterior variances. The method is then further extended to the case where the values of the parameters of the test pieces for measurement are not precisely known. The effectiveness of the proposed methods is demonstrated with academic and real-world physical models.