Solving highdimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
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 FeynmanKac formula in terms of forwardbackward stochastic differential equations. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neuralnetwork 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 FokkerPlanck operator and the linear and nonlinear Schrödinger operators in high dimensions.
 Publication:

Journal of Computational Physics
 Pub Date:
 December 2020
 DOI:
 10.1016/j.jcp.2020.109792
 arXiv:
 arXiv:2002.02600
 Bibcode:
 2020JCoPh.42309792H
 Keywords:

 Diffusion Monte Carlo;
 Deep neural networks;
 Eigenvalue problem;
 Schrödinger equation;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Physics  Computational Physics;
 Statistics  Machine Learning
 EPrint:
 18 pages, 6 figures, 5 tables