论文标题
有限的神经元法和收敛分析
The Finite Neuron Method and Convergence Analysis
论文作者
论文摘要
我们研究了一个基于人工神经网络的$ h^m $ $ $ $ $ - 合格的分段多项式，称为有限的神经元方法（FNM），用于数值解决方案，以$ \ mathbb {r}^d $ $ \ mathbb {r}^d $的数值解决方案，用于任何$ M，d \ geq 1 $，d \ geq 1 $和此方法。给定一个通用域$ω\ subset \ mathbb r^d $和一个分区$ \ mathcal t_h $ of $ω$，在一般中，它仍然是一个开放的问题，如何构造有限的$ h^m（ω）$的符合有限的元素子空间，具有足够的近似属性。通过使用人工神经网络的技术，我们为任何$ k \ ge m $构建了一个由$ h $ $ $ $ $ k $组成的功能组成的函数集的家族，当将它们应用于任何尺寸的任何顺序的椭圆边界值问题时，我们还会获得错误估计。例如，获得了精确解决方案$ u $和有限神经元近似$ u_n $之间的以下错误估计。 $$ \ | u-u_n \ | _ {h^m（ω）} = \ mathcal o（n^{ - {1 \ over 2} - {1 \ over d}}）。 $$的讨论也将在有限神经元方法与有限元方法（FEM）之间的差异和关系上进行。例如，对于有限的神经元方法，基础有限元网格没有先验，并且只能通过求解非线性和非凸优化问题来获得离散解决方案。尽管本文中分析了有限神经元方法的许多理论特性，但其实践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且求解的。为了使读者的完整性和便利性，本手稿中还包括一些基本的已知结果及其证明。
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.
