{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,10,11]],"date-time":"2024-10-11T04:24:09Z","timestamp":1728620649120},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"351","license":[{"start":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T00:00:00Z","timestamp":1749168000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["101077204"],"award-info":[{"award-number":["101077204"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson partial differential equation with Neumann boundary condition. Using Barron\u2019s representation of the solution [IEEE Trans. Inform. Theory 39 (1993), pp.\u00a0930\u2013945] with a probability measure defined on the set of parameter values, the energy is minimized thanks to a gradient curve dynamic on the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2\">\n  <mml:semantics>\n    <mml:mn>2<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-Wasserstein space of the set of parameter values defining the neural network. Inspired by the work from Bach and Chizat [On the global convergence of gradient descent for over-parameterized models using optimal transport, 2018; ICM\u2013International Congress of Mathematicians, EMS Press, Berlin, 2023], we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. Numerical experiments are given to show the potential of the method.<\/p>","DOI":"10.1090\/mcom\/3971","type":"journal-article","created":{"date-parts":[[2024,5,11]],"date-time":"2024-05-11T01:57:37Z","timestamp":1715392657000},"page":"159-208","source":"Crossref","is-referenced-by-count":0,"title":["Numerical solution of Poisson partial differential equation in high dimension using two-layer neural networks"],"prefix":"10.1090","volume":"94","author":[{"given":"Mathias","family":"Dus","sequence":"first","affiliation":[]},{"given":"Virginie","family":"Ehrlacher","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"930","DOI":"10.1109\/18.256500","article-title":"Universal approximation bounds for superpositions of a sigmoidal function","volume":"39","author":"Barron, Andrew R.","year":"1993","journal-title":"IEEE Trans. Inform. 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Yarotsky, Error bounds for approximations with deep ReLU networks, Neural Netw. 94 (2017), 103\u2013114.","DOI":"10.1016\/j.neunet.2017.07.002"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s00365-021-09549-y","article-title":"The Barron space and the flow-induced function spaces for neural network models","volume":"55","author":"E, Weinan","year":"2022","journal-title":"Constr. Approx.","ISSN":"http:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"2","key":"13","doi-asserted-by":"publisher","first-page":"Paper No. 46, 37","DOI":"10.1007\/s00526-021-02156-6","article-title":"Representation formulas and pointwise properties for Barron functions","volume":"61","author":"E., Weinan","year":"2022","journal-title":"Calc. Var. Partial Differential Equations","ISSN":"http:\/\/id.crossref.org\/issn\/0944-2669","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"W. E and S. 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