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

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2020-12-15

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Han, J
Lu, J
Zhou, M

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Abstract

© 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.

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10.1016/j.jcp.2020.109792

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Han, J, J Lu and M Zhou (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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Scholars@Duke

Lu

Jianfeng Lu

James B. Duke Distinguished Professor

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science, machine learning, and other related fields.

More specifically, his current research focuses include:
High dimensional PDEs; generative models and sampling methods; control and reinforcement learning; electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis.


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