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

Accepted

Journal Article

© 2020 Elsevier Inc. 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.

Full Text

Duke Authors

Cited Authors

  • Han, J; Lu, J; Zhou, M

Published Date

  • December 15, 2020

Published In

Volume / Issue

  • 423 /

Electronic International Standard Serial Number (EISSN)

  • 1090-2716

International Standard Serial Number (ISSN)

  • 0021-9991

Digital Object Identifier (DOI)

  • 10.1016/j.jcp.2020.109792

Citation Source

  • Scopus