Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
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© 2020 Elsevier Inc. 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.
Published Version (Please cite this version)10.1016/j.jcp.2020.109792
Publication InfoHan, J; Lu, J; & Zhou, M (2020). Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach. Journal of Computational Physics, 423. pp. 109792-109792. 10.1016/j.jcp.2020.109792. Retrieved from https://hdl.handle.net/10161/21673.
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Professor of Mathematics
Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science and other related fields.More specifically, his current research focuses include:Electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis; rare events and sampling techniques.
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