Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
This work addresses eigenvalue problems in high dimensions, which are computationally challenging in fields like quantum mechanics and statistical physics, but it appears incremental as it builds on existing diffusion Monte Carlo and neural network techniques.
The authors tackled high-dimensional eigenvalue problems for differential operators by proposing a deep neural network method that reformulates the problem as a fixed point of a semigroup flow, using Feynman-Kac formulas and optimization. The method achieved accurate approximations in numerical examples, such as Fokker-Planck and Schrödinger operators, though no specific numerical results or error rates were provided.
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.