The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

E. Weinan; Bing Yu

doi:10.1007/s40304-018-0127-z

The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

E. Weinan, Bing Yu

Mathematics

Research output: Contribution to journal › Article › peer-review

868 Scopus citations

Abstract

We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.

Original language	English (US)
Pages (from-to)	1-12
Number of pages	12
Journal	Communications in Mathematics and Statistics
Volume	6
Issue number	1
DOIs	https://doi.org/10.1007/s40304-018-0127-z
State	Published - Mar 1 2018

All Science Journal Classification (ASJC) codes

Statistics and Probability
Computational Mathematics
Applied Mathematics

Keywords

Deep Ritz Method
Eigenvalue problems
PDE
Variational problems

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10.1007/s40304-018-0127-z

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abstract = "We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.",

keywords = "Deep Ritz Method, Eigenvalue problems, PDE, Variational problems",

author = "E. Weinan and Bing Yu",

note = "Publisher Copyright: {\textcopyright} 2018, School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature.",

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N2 - We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.

AB - We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.

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KW - Eigenvalue problems

KW - PDE

KW - Variational problems

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The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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