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

TL;DR

This paper presents a data-driven approach to learn robust nonlocal models from high-fidelity synthetic data, ensuring well-posedness and robustness even with limited data, applicable to various physical phenomena.

Contribution

It introduces a novel scheme that extracts provably invertible nonlocal models with partially negative kernels, embedding conditions for well-posedness as inequality constraints.

Findings

Successfully reproduces nonlocal kernels from data

Demonstrates numerical homogenization of Darcy flow

Approximates fractional diffusion operators with truncated kernels

Abstract

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.