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On September 7, 2026, a paper titled “Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces” was published under the arXiv cs.LG category (arXiv:2602.19381v2). This study established a regularity theorem for second-order elliptic partial differential equations defined on R^d in spectral Barron spaces. Under mild ellipticity and small-amount assumptions, the study demonstrated that two additional Barron regularities can be obtained. As a consequence, the paper identified a class of PDEs that can be approximated by a two-layer neural network with cosine activation functions, and the width of this network is independent of the spatial dimension.
arXiv:2602.19381v2 发布论文《Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces》，建立了 $\mathbb{R}^{d}$ 上二阶椭圆偏微分方程在谱 Barron 空间中的正则性定理。该研究在 mild ellipticity 和小量假设下，证明解获得两个额外的 Barron 正则度。作为推论，文章识别出一类可由具有余弦激活函数的两层神经网络逼近的 PDE，且其网络宽度与空间维度无关。