巴伦空间中的椭圆正则性理论及其在深度里兹方法中的应用
Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method
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中文总结 AI 辅助
研究巴伦空间中调和函数正则性，证明其通常非利普希茨连续且不在$H^2$等类中，但可被低范数巴伦函数近似。在简单区域成立该结果，并将其应用于深度里兹神经偏微分方程求解器，获先验误差估计。
中文摘要 AI 辅助
我们证明，在巴伦空间（一种为具有单个隐藏层且权重适当有界的宽ReLU网络量身定制的函数类）中具有狄利克雷边界数据的调和函数，通常既不是利普希茨连续的，也不在索伯列夫类$H^2$中。更确切地说，它们不在任何范数控制利普希茨常数的函数类中，这不仅排除了巴伦空间正则性，也排除了具有有界系数的更深ReLU网络在函数类中的正则性。然而，在各种勒贝格和索伯列夫范数（最多有两个导数）中，它们可以被低范数$\sim |\log\varepsilon|$的巴伦函数近似到精度$\sim \varepsilon$。这个积极结果在非常简单的区域成立：任意维度的半空间和二维矩形区域。作为这种正则性理论的应用，我们得到了深度里兹神经偏微分方程求解器的先验误差估计。
英文摘要
We prove that harmonic functions with Dirichlet boundary data in Barron space, a function class tailored to wide ReLU networks with a single hidden layer and suitably bounded weights, are generally neither Lipschitz continuous nor in the Sobolev class $H^2$. A fortiori, they are not in any function class in which the norm controls the Lipschitz constant, which rules out not only Barron space regularity, but also regularity in function classes for deeper ReLU networks with bounded coefficients. They can, however, be approximated to accuracy $\sim \varepsilon$ by Barron functions of low norm $\sim |\log\varepsilon|$ in various Lebesgue and Sobolev norms (with at most two derivatives). The positive result holds on very simple domains: Half-spaces in arbitrary dimension and rectangular domains in two dimensions. As an application of this regularity theory, we obtain a priori error estimates for Deep Ritz neural PDE solvers.
