[2112.14523] Deep neural network approximation theory for high-dimensional functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2112.14523 (math) [Submitted on 29 Dec 2021 ( v1 ), last revised 28 Apr 2026 (this version, v2)] Title: Deep neural network approximation theory for high-dimensional functions Authors: Pierfrancesco Beneventano , Patrick Cheridito , Robin Graeber , Arnulf Jentzen , Benno Kuckuck View a PDF of the paper titled Deep neural network approximation theory for high-dimensional functions, by Pierfrancesco Beneventano and 4 other authors View PDF HTML (experimental) Abstract: The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of dimensionality in the approximation of a large class of functions. More precisely, we prove that these functions can be approximated by DNNs on compact sets such that the number of parameters necessary to represent the approximating DNNs grows at most polynomially in the reciprocal $1/\varepsilon$ of the prescribed approximation error $\varepsilon>0$ and in the input dimension $d\in\mathbb N$. To this end, we introduce certain approximation spaces, consisting of sequences of functions that can be efficiently approximated by DNNs. We then establish closure properties which we combine with known and new bounds on the number of parameters necessary to approximate locally Lipschitz continuous functions, maximum functions, and product functions by DNNs. The main result of this article demonstrates that DNNs have sufficient expressive power to approximate, without the curse of dimensionality, certain sequences of functions which can be constructed by means of a finite number of compositions using locally Lipschitz continuous functions, maxima, and products. Comments: 79 pages, 1 figure Subjects: Numerical Analysis (math.NA) Cite as: arXiv:2112.14523 [math.NA] (or arXiv:2112.14523v2 [math.NA] for this version) https://doi.org/10.48550/arXiv.2112.14523 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Benno Kuckuck [ view email ] [v1] Wed, 29 Dec 2021 12:29:23 UTC (64 KB) [v2] Tue, 28 Apr 2026 23:07:38 UTC (64 KB) Full-text links: Access Paper: View a PDF of the paper titled Deep neural network approximation theory for high-dimensional functions, by Pierfrancesco Beneventano and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2021-12 Change to browse by: cs cs.NA math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from