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

  • Weinan E
  • Bing Yu
Article
  • 237 Downloads

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 Variational problems PDE Eigenvalue problems 

Mathematics Subject Classification

35Q68 

Notes

Acknowledgements

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.

References

  1. 1.
    Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016)MATHGoogle Scholar
  2. 2.
    E, W.: A proposal for machine learning via dynamical systems. Commun. Math. Stat. 5(1), 1–11 (2017)Google Scholar
  3. 3.
    Han, J.Q., Jentzen, A., E, W.: Overcoming the curse of dimensionality: solving high-dimensional partial differential equations using deep learning, submitted, arXiv:1707.02568
  4. 4.
    E, W., Han, J.Q., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, submitted, arXiv:1706.04702
  5. 5.
    Beck, C., E, W., Jentzen, A.: Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, submitted. arXiv:1709.05963
  6. 6.
    Han, J.Q., Zhang, L., Car, R., E, W.: Deep potential: a general and “first-principle” representation of the potential energy, submitted, arXiv:1707.01478
  7. 7.
    Zhang, L., Han, J.Q., Wang, H., Car, R., E, W.: Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics, submitted, arXiv:1707.09571
  8. 8.
    Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)MATHGoogle Scholar
  9. 9.
    He, K.M., Zhang, X.Y., Ren, S.Q., Sun, J.: Deep residual learning for image recognition. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778 (2016).  https://doi.org/10.1109/CVPR.2016.90
  10. 10.
    Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. arXiv preprint. arXiv:1412.6980, (2014)
  11. 11.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method. Prentice-Hall, Upper Saddle River (1973)MATHGoogle Scholar
  12. 12.
    Huang, G., Liu, Z., Weinberger, K.Q., Laurens, V.D.M.: Densely connected convolutional networks. arXiv preprint. arXiv:1608.06993, (2016)

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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