Kernel Methods in the Deep Ritz framework: Theory and practice

Kleikamp, Hendrik; Wenzel, Tizian

Forschungsartikel in Online-Sammlung | Preprint | Peer reviewed

Zusammenfassung

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche’s method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated numerically and the results agree with the theoretical findings.

Details zur Publikation

Name des RepositoriumsarXiv
Artikelnummer2410.03503
Statuseingereicht / in Begutachtung
Veröffentlichungsjahr2024 (07.10.2024)
Sprache, in der die Publikation verfasst istEnglisch
DOI10.48550/arXiv.2410.03503
Link zum Volltexthttps://doi.org/10.48550/arXiv.2410.03503
StichwörterDeep Ritz method; kernel methods; energy minimization; partial differential equations; Nitsche’s method; a priori error estimation

Autor*innen der Universität Münster

Kleikamp, Hendrik
Professur für Angewandte Mathematik, insbesondere Numerik (Prof. Ohlberger)