TY - JOUR
T1 - Solving high-dimensional eigenvalue problems using deep neural networks
T2 - A diffusion Monte Carlo like approach
AU - Han, Jiequn
AU - Lu, Jianfeng
AU - Zhou, Mo
N1 - Funding Information:
The work of JL and MZ is supported in part by National Science Foundation via grant DMS-1454939 and DMS-2012286. The authors are grateful for computing time at the Terascale Infrastructure for Groundbreaking Research in Science and Engineering (TIGRESS) of Princeton University.
Funding Information:
The work of JL and MZ is supported in part by National Science Foundation via grant DMS-1454939 and DMS-2012286 . The authors are grateful for computing time at the Terascale Infrastructure for Groundbreaking Research in Science and Engineering (TIGRESS) of Princeton University.
Publisher Copyright:
© 2020 Elsevier Inc.
PY - 2020/12/15
Y1 - 2020/12/15
N2 - 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.
AB - 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.
KW - Deep neural networks
KW - Diffusion Monte Carlo
KW - Eigenvalue problem
KW - Schrödinger equation
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U2 - 10.1016/j.jcp.2020.109792
DO - 10.1016/j.jcp.2020.109792
M3 - Article
AN - SCOPUS:85091906214
SN - 0021-9991
VL - 423
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 109792
ER -