Fréchet度量下的逼近速率：Barron空间、Paley-Wiener空间与傅里叶乘子
Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers
AI总结：
该研究针对偏微分方程算子学习的逼近问题，在Fréchet度量下分析线性微分算子的傅里叶域符号逼近能力，给出误差可达的充分条件并推广定理，还结合指数谱Barron空间结果给出可良好逼近的符号实例。
摘要：
算子学习是借助神经网络模拟偏微分方程（PDE）领域的一项新近进展。该方法的核心思路是学习算子的行为，使得到的神经网络成为无穷维空间中的一个（近似）映射，能够（近似）模拟由PDE支配的解算子。在本研究中，我们通过在傅里叶域中逼近相应的符号，研究了线性微分算子的若干通用逼近能力。类比Hörmander符号类的结构，我们考虑由一列半范数诱导的拓扑下的逼近。在此框架下，我们采用Fréchet度量来衡量逼近误差，且我们的主要结果确定了达到预设逼近误差的充分条件。其次，我们进一步聚焦于主定理的一个自然推广，在该推广中我们成功弱化了对这列半范数的假设。基于指数谱Barron空间的现有逼近结果，我们随后给出了一类可被良好逼近的符号的具体示例。
英文摘要：
Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network is an (approximate) mapping in infinite-dimensional spaces that is capable of (approximately) simulating the solution operator governed by the PDE. In our work, we study some general approximation capabilities for linear differential operators by approximating the corresponding symbol in the Fourier domain. Analogous to the structure of the class of Hörmander-Symbols, we consider the approximation with respect to a topology that is induced by a sequence of semi-norms. In that sense, we measure the approximation error in terms of a Fréchet metric, and our main result identifies sufficient conditions for achieving a predefined approximation error. Secondly, we then focus on a natural extension of our main theorem, in which we manage to reduce the assumptions on the sequence of semi-norms. Based on existing approximation results for the exponential spectral Barron space, we then present a concrete example of symbols that can be approximated well.
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