%0 Generic %1 han2020solving %A Han, Jiequn %A Lu, Jianfeng %A Zhou, Mo %D 2020 %K quantumcomputing %R 10.1016/j.jcp.2020.109792 %T Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach %U http://arxiv.org/abs/2002.02600 %X 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.

@misc{han2020solving, 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 Schr\"odinger operators in high dimensions.}, added-at = {2023-01-21T16:00:02.000+0100}, author = {Han, Jiequn and Lu, Jianfeng and Zhou, Mo}, biburl = {https://www.bibsonomy.org/bibtex/29bfbd539eddf2da3465a91a4b2c5eab3/cmcneile}, description = {Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach}, doi = {10.1016/j.jcp.2020.109792}, interhash = {01d2409df70ae0acbb49b8a137ec1c50}, intrahash = {9bfbd539eddf2da3465a91a4b2c5eab3}, keywords = {quantumcomputing}, note = {cite arxiv:2002.02600Comment: 18 pages, 6 figures, 5 tables}, timestamp = {2023-01-21T16:00:02.000+0100}, title = {Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach}, url = {http://arxiv.org/abs/2002.02600}, year = 2020 }