  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 Schr\"odinger operators in high dimensions.
