Neural and spectral operator surrogates

• Verfasst von: Herrmann, Lukas [VerfasserIn]
• Verfasst von: Schwab, Christoph [VerfasserIn]
• Verfasst von: Zech, Jakob [VerfasserIn]
• Titel: Neural and spectral operator surrogates
• Titelzusatz: unified construction and expression rate bounds
• Verf.angabe: Lukas Herrmann, Christoph Schwab, Jakob Zech
• E-Jahr: 2024
• Jahr: 15 July 2024
• Umfang: 43 S.
• Fussnoten: Gesehen am 13.01.2025
• Titel Quelle: Enthalten in: Advances in computational mathematics
• Ort Quelle: Bussum : Baltzer Science Publ., 1993
• Jahr Quelle: 2024
• Band/Heft Quelle: 50(2024), 4, Artikel-ID 72, Seite 1-43
• ISSN Quelle: 1572-9044
• DOI: doi:10.1007/s10444-024-10171-2
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 35J15
• Sach-SW: 41A25
• Sach-SW: 65D40
• Sach-SW: 65N35
• Sach-SW: 68T07
• Sach-SW: Generalized polynomial chaos
• Sach-SW: Neural networks
• Sach-SW: Operator learning
• K10plus-PPN: 1914432061

Abstract

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising, e.g., as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for deep neural operator and generalized polynomial chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases, or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus.