Data-driven Whitney forms for structure-preserving control volume analysis

Journal Article · 16 October 2023 · Journal of Computational Physics

DOI: https://doi.org/10.1016/j.jcp.2023.112520 · OSTI ID:2311272

Actor, Jonas A.  ^[1]; Hu, Xiaozhe ^[2]; Huang, Andy ^[3]; Roberts, Scott Alan ^[4]; Trask, Nathaniel ^[1]

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1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
2. Tufts Univ., Medford, MA (United States)
3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
4. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Engineering Sciences Center

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $$\mathbb{R}$$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

• Research Organization: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: NA0003525
• OSTI ID: 2311272
• Report Number(s): SAND-2023-13527J; TRN: US2409331
• Journal Information: Journal of Computational Physics, Vol. 496; ISSN 0021-9991
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English