arXiv:2602.19381math.APcs.LG2026-02被引 2
证明了椭圆型方程解在谱Barron空间中的光滑性提升。
Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces
- 在弱椭圆性和小参数条件下，解的Barron正则性提升两个阶数。
- 解可用两层余弦激活神经网络逼近，宽度与维度无关。
- 适用于高维问题的神经网络近似，理论支持强。
我们建立了 $ ^{d}$ 上二阶椭圆型偏微分方程在谱Barron空间中的正则性定理。在温和的椭圆性及小参数假设下，解的Barron正则性可提升两个阶数。作为推论，我们识别出一类解能被两层余弦激活神经网络逼近的PDE，且网络宽度不随空间维度增加而增长。
原文摘要 · Abstract (English)
We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.
偏微分方程神经网络逼近正则性理论
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