Kernel Manifolds: Nonlinear‐Augmentation Dimensionality Reduction Using Reproducing Kernel Hilbert Spaces

Diaz, Alejandro N.; Needels, Jacob Troy; Tezaur, Irina Kalashnikova; Blonigan, Patrick Joseph

doi:10.1002/nme.70230

Kernel Manifolds: Nonlinear‐Augmentation Dimensionality Reduction Using Reproducing Kernel Hilbert Spaces

Journal Article · Mon Dec 22 00:00:00 EST 2025 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.70230· OSTI ID:3018772

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Sandia National Lab. (SNL-CA), Livermore, CA (United States)

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. In conclusion, we compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: NA0003525

Other Award/Contract Number:: 22025291

OSTI ID:: 3018772

Report Number(s):: SAND--2026-16616J; 1787910

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering Journal Issue: 24 Vol. 126; ISSN 0029-5981; ISSN 1097-0207

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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