Solving high-dimensional 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 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.
- 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
- E-Print:
- 18 pages, 6 figures, 5 tables