人工神经网络的高维近似空间及其在偏微分方程中的应用
High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations
AI总结：
本文提出人工神经网络近似空间的概念及相关工具，证明其在多种操作下封闭，并运用该理论证明ANN在数值近似一阶输运偏微分方程时能克服维数灾难。
摘要：
在本文中，我们开发了一种新的机制来研究人工神经网络（ANN）在不遭受维数灾难的情况下近似高维函数的能力。具体而言，我们引入了一个被称为人工神经网络近似空间的概念，并提出了处理这些空间的若干工具。粗略地说，近似空间由函数序列组成，这些函数能以合适的方式被ANN无维数灾难地近似，即以精度$\varepsilon > 0$近似该序列中的函数所需的ANN参数数量，至多以所需精度的倒数$1/\varepsilon$和函数的维度$d \in \mathbb{N} = \{1, 2, 3, \ldots \}$呈多项式增长。我们证明这些近似空间在包括线性组合、极限形成和无限复合在内的多种操作下是封闭的。为了说明本文提出机制的实用性，我们运用所发展的理论证明，ANN在数值近似某些一阶输运偏微分方程（PDE）时具备克服维数灾难的能力。我们甚至证明，近似空间在一阶输运PDE的流作用下是封闭的。
英文摘要：
In this paper we develop a new machinery to study the capacity of artificial neural networks (ANNs) to approximate high-dimensional functions without suffering from the curse of dimensionality. Specifically, we introduce a concept which we refer to as approximation spaces of artificial neural networks and we present several tools to handle those spaces. Roughly speaking, approximation spaces consist of sequences of functions which can, in a suitable way, be approximated by ANNs without curse of dimensionality in the sense that the number of required ANN parameters to approximate a function of the sequence with an accuracy $\varepsilon > 0$ grows at most polynomially both in the reciprocal $1/\varepsilon$ of the required accuracy and in the dimension $d \in \mathbb{N} = \{1, 2, 3, \ldots \}$ of the function. We show that these approximation spaces are closed under various operations including linear combinations, formations of limits, and infinite compositions. To illustrate the utility of the machinery proposed in this paper, we employ the developed theory to prove that ANNs have the capacity to overcome the curse of dimensionality in the numerical approximation of certain first order transport partial differential equations (PDEs). We even prove that approximation spaces are closed under flows of first order transport PDEs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
