Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory

• URL: http://arxiv.org/abs/2006.15733v2
• Date: Thu, 10 Dec 2020 20:43:11 GMT
• Title: Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory
• Authors: Tao Luo and Haizhao Yang
• Abstract summary: We show that the gradient descent method can identify a global minimizer of the least-squares optimization for solving second-order linear PDEs. We also analyze the generalization error of the least-squares optimization for second-order linear PDEs and two-layer neural networks.
• Score: 4.243322291023028
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The problem of solving partial differential equations (PDEs) can be formulated into a least-squares minimization problem, where neural networks are used to parametrize PDE solutions. A global minimizer corresponds to a neural network that solves the given PDE. In this paper, we show that the gradient descent method can identify a global minimizer of the least-squares optimization for solving second-order linear PDEs with two-layer neural networks under the assumption of over-parametrization. We also analyze the generalization error of the least-squares optimization for second-order linear PDEs and two-layer neural networks, when the right-hand-side function of the PDE is in a Barron-type space and the least-squares optimization is regularized with a Barron-type norm, without the over-parametrization assumption.

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