Bayesian Nonlocal Operator Regression: A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification

We consider the problem of modeling heterogeneous materials where microscale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, additive independent identically distributed Gaussian noise is employed to model the discrepancy between the nonlocal model and the data. Then, we use a Markov chain Monte Carlo (MCMC) method to sample the posterior probability distribution on parameters involved in the nonlocal constitutive law and associated modeling discrepancies relative to higher-fidelity computations. As an application, we consider the propagation of stress waves through a one-dimensional heterogeneous bar with randomly generated microstructure. Several numerical tests illustrate the construction, enabling UQ in nonlocal model predictions. Although nonlocal models have become popular means for homogenization, their statistical calibration with respect to high-fidelity models has not been presented before. This work is a first step in this direction, focused on Bayesian parameter calibration.