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

  • Weinan E
  • Bing Yu


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.


Deep Ritz Method Variational problems PDE Eigenvalue problems 

Mathematics Subject Classification




We are grateful to Professor Ruo Li and Dr. Zhanxing Zhu for very helpful discussions. The work of E and Yu is supported in part by the National Key Basic Research Program of China 2015CB856000, Major Program of NNSFC under Grant 91130005, DOE Grant DE-SC0009248, and ONR Grant N00014-13-1-0338.


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Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Beijing Institute of Big Data ResearchBeijingChina
  2. 2.Department of Mathematics and PACMPrinceton UniversityPrincetonUSA
  3. 3.School of Mathematical Sciences and BICMRPeking UniversityBeijingChina
  4. 4.School of Mathematical SciencesPeking UniversityBeijingChina

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