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

• URL: http://arxiv.org/abs/2002.02600v2
• Date: Thu, 16 Jul 2020 03:34:03 GMT
• Title: Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
• Authors: Jiequn Han, Jianfeng Lu, Mo Zhou
• Abstract summary: The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples.
• Score: 14.558626910178127
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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