Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory
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.
Cite
@article{arxiv.2006.15733,
title = {Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory},
author = {Tao Luo and Haizhao Yang},
journal= {arXiv preprint arXiv:2006.15733},
year = {2020}
}