Uniform convergence guarantees for the deep Ritz method for nonlinear problems

被引：5

作者：

Dondl, Patrick ^{[1
]}

Mueller, Johannes ^{[2
]}

Zeinhofer, Marius ^{[3
]}

机构：

[1] Univ Freiburg, Dept Appl Math, Hermann Herder Str 10, D-79104 Freiburg, Germany

[2] Max Planck Inst Math Sci, Inselstr 22, D-04103 Leipzig, Germany

[3] Simula Res Lab, Dept Numer Anal & Sci Comp, Kristian Augusts Gate 23, N-0164 Oslo, Norway

来源：

ADVANCES IN CONTINUOUS AND DISCRETE MODELS | 2022年 / 2022卷 / 01期

基金：

欧盟地平线“2020”; 欧洲研究理事会;

关键词：

Calculus of variations; Nonlinear problems; Ritz method; Boundary penalty method; Neural networks; ALGORITHM;

D O I：

10.1186/s13662-022-03722-8

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页数：19

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