On the Representation of Solutions to Elliptic PDEs in Barron Spaces
On the Representation of Solutions to Elliptic PDEs in Barron Spaces
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发表时间：
2021-06
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通讯作者：
Ziang Chen;Jianfeng Lu;Yulong Lu
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作者：
Ziang Chen;Jianfeng Lu;Yulong Lu
基于神经网络的高维偏微分方程数值解已经取得了令人振奋的进展。本文在巴伦空间中导出了d维二阶椭圆型偏微分方程解的复杂性估计。巴伦空间是一组函数，允许某个参数岭函数对参数上的概率测度进行积分。在适当的假设下，证明了若椭圆型偏微分方程的系数和源项都位于巴伦空间中，则方程的解关于H范数-接近于一个巴伦函数.此外，我们证明dimensionexplicit界的巴伦规范的近似解，取决于最多多项式的PDE的尺寸D。作为一个直接后果的复杂性估计，PDE的解决方案可以近似在任何有界域上的两层神经网络的H范数与维数显式收敛速度。
Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is -close with respect to the H norm to a Barron function. Moreover, we prove dimensionexplicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H norm with a dimension-explicit convergence rate.