{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/deep-nitsche-method-deep-ritz-method-with","title":"Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions","arxiv_id":"1912.01309","date":"2019-12-03","proceeding":null,"authors":["Yulei Liao","Pingbing Ming"],"abstract":"We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsche's variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs. We prove the error estimate in the energy norm and illustrate the method on several representative problems posed in at most 100 dimension.","url_abs":"https://arxiv.org/abs/1912.01309v1","url_pdf":"https://arxiv.org/pdf/1912.01309v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"deep-nitsche-method-deep-ritz-method-with","repo_url":"https://github.com/liaoyulei/DeepNitscheMethod","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"tf","reach":{"status":"ok"}},{"paper_slug":"deep-nitsche-method-deep-ritz-method-with","repo_url":"https://github.com/junbinhuang/DeepRitz","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1912.01309","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}