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Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

Publication ,  Journal Article
Han, J; Lu, J; Zhou, M
Published in: Journal of Computational Physics
December 15, 2020

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman-Kac formula in terms of forward-backward stochastic differential equations. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. The criterion of fixed point provides a natural loss function to search for parameters via optimization. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples, including Fokker-Planck operator and the linear and nonlinear Schrödinger operators in high dimensions.

Duke Scholars

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Published In

Journal of Computational Physics

DOI

EISSN

1090-2716

ISSN

0021-9991

Publication Date

December 15, 2020

Volume

423

Related Subject Headings

  • Applied Mathematics
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences
 

Citation

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Han, J., Lu, J., & Zhou, M. (2020). Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach. Journal of Computational Physics, 423. https://doi.org/10.1016/j.jcp.2020.109792
Han, J., J. Lu, and M. Zhou. “Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach.” Journal of Computational Physics 423 (December 15, 2020). https://doi.org/10.1016/j.jcp.2020.109792.
Han, J., et al. “Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach.” Journal of Computational Physics, vol. 423, Dec. 2020. Scopus, doi:10.1016/j.jcp.2020.109792.
Journal cover image

Published In

Journal of Computational Physics

DOI

EISSN

1090-2716

ISSN

0021-9991

Publication Date

December 15, 2020

Volume

423

Related Subject Headings

  • Applied Mathematics
  • 51 Physical sciences
  • 49 Mathematical sciences
  • 40 Engineering
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences