{"title":"非线性问题的深里兹方法的一致收敛保证。","authors":"Patrick Dondl, Johannes Müller, Marius Zeinhofer","doi":"10.1186/s13662-022-03722-8","DOIUrl":null,"url":null,"abstract":"<p><p>We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the <i>p</i>-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides.</p>","PeriodicalId":72091,"journal":{"name":"Advances in continuous and discrete models","volume":null,"pages":null},"PeriodicalIF":2.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9287260/pdf/","citationCount":"7","resultStr":"{\"title\":\"Uniform convergence guarantees for the deep Ritz method for nonlinear problems.\",\"authors\":\"Patrick Dondl, Johannes Müller, Marius Zeinhofer\",\"doi\":\"10.1186/s13662-022-03722-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the <i>p</i>-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides.</p>\",\"PeriodicalId\":72091,\"journal\":{\"name\":\"Advances in continuous and discrete models\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9287260/pdf/\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in continuous and discrete models\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1186/s13662-022-03722-8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2022/7/15 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in continuous and discrete models","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1186/s13662-022-03722-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2022/7/15 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Uniform convergence guarantees for the deep Ritz method for nonlinear problems.
We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides.
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