We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\mathrm{D} \subset \mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\mathrm{D}$, comprising the countably-normed spaces of I.M. Babu\v{s}ka and B.Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial high order (``$p$-version'') finite elements with elementwise polynomial degree $p\in\mathbb{N}$ on arbitrary, regular, simplicial partitions of polyhedral domains $\mathrm{D} \subset \mathbb{R}^d$, $d\geq 2$ can be exactly emulated by neural networks combining ReLU and ReLU$^2$ activations. On shape-regular, simplicial partitions of polytopal domains $\mathrm{D}$, both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the finite element space, in particular for the $hp$-Finite Element Method of I.M. Babu\v{s}ka and B.Q. Guo.

翻译：我们分析了在有限多面体区域$\mathrm{D} \subset \mathbb{R}^d$（$d=2,3$）中，深度神经网络对含点奇点光滑函数的逼近率。针对由$\mathrm{D}$中加权Sobolev尺度定义的Gevrey正则解类（包括I.M. Babu\v{s}ka和B.Q. Guo的可数赋范空间），我们证明了在神经元数量和基于非零系数数量的Sobolev空间指数级逼近率。作为中间结果，我们证明了对于$\mathbb{R}^d$（$d\geq 2$）中多面体区域$\mathrm{D}$的任意正则单纯形剖分，阶数为$p\in\mathbb{N}$的连续分片多项式高阶（"$p$版本"）有限元可通过结合ReLU和ReLU$^2$激活函数的神经网络精确模拟。在$\mathrm{D}$的形状正则单纯形剖分上，神经元数量和非零参数数量均与有限元空间的自由度数量成正比，尤其适用于I.M. Babu\v{s}ka和B.Q. Guo提出的$hp$有限元方法。