{"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/nonclosedness-of-the-set-of-neural-networks","title":"Nonclosedness of Sets of Neural Networks in Sobolev Spaces","arxiv_id":"2007.11730","date":"2020-07-23","proceeding":null,"authors":["Scott Mahan","Emily King","Alex Cloninger"],"abstract":"We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly $m$-times differentiable activation function $\\rho$, we construct a sequence of neural networks $(\\Phi_n)_{n \\in \\mathbb{N}}$ whose realizations converge in order-$(m-1)$ Sobolev norm to a function that cannot be realized exactly by a neural network. Thus, sets of realized neural networks are not closed in order-$(m-1)$ Sobolev spaces $W^{m-1,p}$ for $p \\in [1,\\infty]$. We further show that these sets are not closed in $W^{m,p}$ under slightly stronger conditions on the $m$-th derivative of $\\rho$. For a real analytic activation function, we show that sets of realized neural networks are not closed in $W^{k,p}$ for any $k \\in \\mathbb{N}$. The nonclosedness allows for approximation of non-network target functions with unbounded parameter growth. We partially characterize the rate of parameter growth for most activation functions by showing that a specific sequence of realized neural networks can approximate the activation function's derivative with weights increasing inversely proportional to the $L^p$ approximation error. Finally, we present experimental results showing that networks are capable of closely approximating non-network target functions with increasing parameters via training.","url_abs":"https://arxiv.org/abs/2007.11730v4","url_pdf":"https://arxiv.org/pdf/2007.11730v4.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":"abstracts"},"code_links":[{"paper_slug":"nonclosedness-of-the-set-of-neural-networks","repo_url":"https://github.com/scmahan/Neural-Network-Sobolev-Training","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}