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

arXiv: https://arxiv.org/abs/2211.01330 · HTML: https://mlsanity.com/paper/2211.01330/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 63 references that are in the map.
- First posted on arXiv 6 Oct 2022.

## Builds on (most cited first, 10 of 10)

- [Generalized Multiscale Finite Element Methods (GMsFEM)](https://mlsanity.com/paper/1301.2866/) (2013)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [A data-driven peridynamic continuum model for upscaling molecular dynamics](https://mlsanity.com/paper/2108.04883/) (2021)
- [Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws](https://mlsanity.com/paper/2012.04157/) (2020)
- [Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction](https://mlsanity.com/paper/1510.06053/) (2015)
- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)
- [Embedded Model Error Representation for Bayesian Model Calibration](https://mlsanity.com/paper/1801.06768/) (2018)
- [Data-driven learning of robust nonlocal physics from high-fidelity synthetic data](https://mlsanity.com/paper/2005.10076/) (2020)
- [MetaNOR: A Meta-Learnt Nonlocal Operator Regression Approach for Metamaterial Modeling](https://mlsanity.com/paper/2206.02040/) (2022)
- [Deriving peridynamic influence functions for one-dimensional elastic materials with periodic microstructure](https://mlsanity.com/paper/2003.05520/) (2020)

## Built on by (most cited first, 3 of 3)

- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach](https://mlsanity.com/paper/2409.00220/) (2024)
- [Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators](https://mlsanity.com/paper/2410.20331/) (2024)

## Abstract (arXiv)

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.