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

Journal Article · Mon Dec 07 00:00:00 EST 2020 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2020.113553· OSTI ID:1765754

You, Huaiqian ^[1]; Yu, Yue ^[1]; Trask, Nathaniel ^[2]; Gulian, Mamikon ^[2]; D’Elia, Marta ^[3]

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1. Lehigh Univ., Bethlehem, PA (United States)
2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
3. Sandia National Lab. (SNL-CA), Livermore, CA (United States)

A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. Here, we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity — it is particularly challenging to extract surrogates which are provably well-posed and numerically stable. Our scheme not only yields a convex optimization problem, but also allows extraction of nonlocal models whose kernels may be partially negative while maintaining well-posedness even in small-data regimes. 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 to meet the requisite conditions of the nonlocal theory. 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.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Sandia National Laboratories, Livermore, CA
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
• Grant/Contract Number: AC04-94AL85000; NA0003525
• OSTI ID: 1765754
• Alternate ID(s): OSTI ID: 1809475
• Report Number(s): SAND--2020-5158J; 686140
• Journal Information: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 374; ISSN 0045-7825
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
data-driven learning
fractional models
homgenization
machine learning
multiscale modeling
nonlocal models
operator regression
optimization