有限神经元方法及其收敛性分析
The Finite Neuron Method and Convergence Analysis
摘要：
研究了一类基于人工神经网络的$H^m$ -协调分段多项式，即有限神经元方法（FNM），用于求解任意$m,d \geq 1$的$\mathbb{R}^d$中的$H^m$-阶偏微分方程，并给出了该方法的收敛性分析.给定一般域$Ω\subset\mathbb R^d$和$Ω$的一个划分$\mathcal T_h$，如何构造$H^m(Ω)$的协调有限元子空间，使其具有足够的逼近性质，在一般情况下仍是一个未解决的问题.利用人工神经网络的技巧，对任意$k\ge m$，构造了一族由$k$次分段多项式组成的$H^m$ -相容函数集，并进一步得到了它们用于求解任意维任意阶椭圆边值问题的误差估计.例如，获得了精确解$u$和有限神经元逼近$u_N$之间的以下误差估计。$$ \|u-u_N\|_{H^m(Ω)}=\mathcal O(N^{-{1\over 2}-{1\over d}}). $$还讨论了有限神经元方法和有限元方法之间的区别和联系。例如，对于有限神经元方法，底层有限元网格不是先验给定的，并且离散解只能通过求解非线性和非凸优化问题来获得。尽管本文中分析的有限神经元方法具有许多理想的理论性质，但其实用价值是进一步研究的主题，因为上述潜在的非线性和非凸优化问题可能是昂贵的且具有挑战性的。为了完整和方便读者，本文还包括了一些基本的已知结果及其证明。
英文摘要：
We study a family of $H^m$-conforming piecewise polynomials based on artificial neural network, named as the finite neuron method (FNM), for numerical solution of $2m$-th order partial differential equations in $\mathbb{R}^d$ for any $m,d \geq 1$ and then provide convergence analysis for this method. Given a general domain $Ω\subset\mathbb R^d$ and a partition $\mathcal T_h$ of $Ω$, it is still an open problem in general how to construct conforming finite element subspace of $H^m(Ω)$ that have adequate approximation properties. By using techniques from artificial neural networks, we construct a family of $H^m$-conforming set of functions consisting of piecewise polynomials of degree $k$ for any $k\ge m$ and we further obtain the error estimate when they are applied to solve elliptic boundary value problem of any order in any dimension. For example, the following error estimates between the exact solution $u$ and finite neuron approximation $u_N$ are obtained. $$ \|u-u_N\|_{H^m(Ω)}=\mathcal O(N^{-{1\over 2}-{1\over d}}). $$ Discussions will also be given on the difference and relationship between the finite neuron method and finite element methods (FEM). For example, for finite neuron method, the underlying finite element grids are not given a priori and the discrete solution can only be obtained by solving a non-linear and non-convex optimization problem. Despite of many desirable theoretical properties of the finite neuron method analyzed in the paper, its practical value is a subject of further investigation since the aforementioned underlying non-linear and non-convex optimization problem can be expensive and challenging to solve. For completeness and also convenience to readers, some basic known results and their proofs are also included in this manuscript.
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