Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators

TL;DR

This paper introduces ENOR, a novel framework that learns nonlocal homogenized models and quantifies their structural errors, improving long-term predictions of wave propagation in heterogeneous materials.

Contribution

ENOR combines nonlocal operator regression with Bayesian inference and multilevel MCMC to accurately learn and quantify model errors in nonlocal homogenization.

Findings

ENOR outperforms additive noise models in uncertainty estimation.

The framework effectively captures structural model errors in long-term simulations.

Application to wave propagation demonstrates improved predictive reliability.

Abstract

Nonlocal, integral operators have become an efficient surrogate for bottom-up homogenization, due to their ability to represent long-range dependence and multiscale effects. However, the nonlocal homogenized model has unavoidable discrepancy from the microscale model. Such errors accumulate and propagate in long-term simulations, making the resultant prediction unreliable. To develop a robust and reliable bottom-up homogenization framework, we propose a new framework, which we coin Embedded Nonlocal Operator Regression (ENOR), to learn a nonlocal homogenized surrogate model and its structural model error. This framework provides discrepancy-adaptive uncertainty quantification for homogenized material response predictions in long-term simulations. The method is built on Nonlocal Operator Regression (NOR), an optimization-based nonlocal kernel learning approach, together with an embedded model error term in the trainable kernel. Then, Bayesian inference is employed to infer the model error term parameters together with the kernel parameters. To make the problem computationally feasible, we use a multilevel delayed acceptance Markov chain Monte Carlo (MLDA-MCMC) method, enabling efficient Bayesian model calibration and model error estimation. We apply this technique to predict long-term wave propagation in a heterogeneous one-dimensional bar, and compare its performance with additive noise models. Owing to its ability to capture model error, the learned ENOR achieves improved estimation of posterior predictive uncertainty.