周期函数的谱Barron空间
Spectral Barron spaces of periodic functions
- Université de Lorraine（洛林大学）
- Fudan University（复旦大学）
- Okayama University（冈山大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究周期函数的谱Barron空间，将其推广到高维环面和局部紧阿贝尔群，通过傅里叶级数与加权序列空间等距同构，并探讨了PDE适定性和两个反问题的条件稳定性。
AI中文摘要：
谱Barron空间是傅里叶-勒贝格型空间，出现在神经网络逼近函数的研究中。常规的谱Barron空间主要在全空间或具有边界光滑性假设的有界域上考虑。本文研究了周期函数的谱Barron空间，从而将谱Barron空间的研究推广到高维环面，更一般地，推广到局部紧阿贝尔群。这些推广通过考虑傅里叶级数展开来实现，该展开与基于$\ell^1$序列空间的加权序列空间等距同构。我们讨论了这些新引入空间中偏微分方程的适定性。研究了两个具体的演化方程反问题，在环面上的谱Barron空间框架内获得了条件稳定性。
英文摘要：
Spectral Barron spaces are Fourier-Lebesgue-type spaces that arise in the approximation of functions by neural networks. Conventional spectral Barron spaces are mostly considered on the whole space or on a bounded domain with certain smoothness assumptions on the boundary. In this work, we investigate spectral Barron spaces for periodic functions, thereby extending the study of spectral Barron spaces to high-dimensional tori and, more generally, to locally compact Abelian groups. These extensions are carried out by considering Fourier series expansions that are isometrically isomorphic to weighted sequence spaces built upon the $\ell^1$ sequence space. We discuss the well-posedness of PDEs in these newly introduced spaces. Two specific inverse problems for evolution equations are investigated, yielding conditional stability within the framework of spectral Barron spaces on the torus.
