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

Jiequn Han, Jianfeng Lu, Mo Zhou
2020 arXiv   pre-print
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
more » ... e 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 Schr\"odinger operators in high dimensions.
arXiv:2002.02600v2 fatcat:v3kncuqls5bulht7aq6zvsdl7y