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

Jiequn Han, Jianfeng Lu, Mo Zhou

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Article number109792
JournalJournal of Computational Physics
Volume423
DOIs
StatePublished - Dec 15 2020

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Deep neural networks
  • Diffusion Monte Carlo
  • Eigenvalue problem
  • Schrödinger equation

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