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

TL;DR

This paper introduces a Bayesian framework combining nonlocal operator regression and Bayesian inference to quantify uncertainty in modeling heterogeneous materials, validated through stress wave propagation in a microstructured bar.

Contribution

It develops the first Bayesian approach for statistical calibration and uncertainty quantification of nonlocal models in material homogenization.

Findings

Successful UQ in nonlocal model predictions demonstrated

Bayesian calibration improves model reliability

Framework applicable to complex heterogeneous materials

Abstract

We consider the problem of modeling heterogeneous materials where micro-scale 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, 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 towards statistical characterization of nonlocal model discrepancy in the context of homogenization.