{"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/a-deep-fourier-residual-method-for-solving","title":"A Deep Fourier Residual Method for solving PDEs using Neural Networks","arxiv_id":"2210.14129","date":"2022-10-25","proceeding":null,"authors":["Jamie M. Taylor","David Pardo","Ignacio Muga"],"abstract":"When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides with--or is equivalent to--the $H^{-1}$-norm of the residual; however, it is often difficult to accurately compute it. This work assumes rectangular domains and proposes the use of a Discrete Sine/Cosine Transform to accurately and efficiently compute the $H^{-1}$ norm. The resulting Deep Fourier-based Residual (DFR) method efficiently and accurately approximate solutions to PDEs. This is particularly useful when solutions lack $H^{2}$ regularity and methods involving strong formulations of the PDE fail. We observe that the $H^1$-error is highly correlated with the discretised loss during training, which permits accurate error estimation via the loss.","url_abs":"https://arxiv.org/abs/2210.14129v1","url_pdf":"https://arxiv.org/pdf/2210.14129v1.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":"a-deep-fourier-residual-method-for-solving","repo_url":"https://github.com/mathmode/adaptive-deep-fourier-residual-method-via-overlapping-domain-decomposition","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}