# Data-driven learning of robust nonlocal physics from high-fidelity synthetic data

arXiv: https://arxiv.org/abs/2005.10076 · HTML: https://mlsanity.com/paper/2005.10076/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2020.

## Built on by (most cited first, 6 of 6)

- [A data-driven peridynamic continuum model for upscaling molecular dynamics](https://mlsanity.com/paper/2108.04883/) (2021)
- [Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks](https://mlsanity.com/paper/2201.09989/) (2022)
- [MetaNOR: A Meta-Learnt Nonlocal Operator Regression Approach for Metamaterial Modeling](https://mlsanity.com/paper/2206.02040/) (2022)
- [Machine Learning in Heterogeneous Porous Materials](https://mlsanity.com/paper/2202.04137/) (2022)
- [Bayesian Nonlocal Operator Regression (BNOR): A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification](https://mlsanity.com/paper/2211.01330/) (2022)
- [Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures](https://mlsanity.com/paper/2301.04540/) (2023)

## Abstract (arXiv)

A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting provably robust data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity. Our scheme allows extraction of provably invertible nonlocal models whose kernels may be partially negative. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints and ensure that models are robust, even in small-data regimes. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels.