This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+1\}\big)$ and depth $\mathcal{O}(L)$ can approximate an arbitrary H\"older continuous function of order $\alpha\in (0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}\big(\sqrt{d} N^{-2\alpha/d}L^{-2\alpha/d}\big)$ measured in $L^p$-norm for any $N,L\in \mathbb{N}^+$ and $p\in[1,\infty]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $\omega_f(\cdot)$, the constructive approximation rate is $\mathcal{O}\big(\sqrt{d}\,\omega_f( N^{-2/d}L^{-2/d})\big)$. We also extend our analysis to $f$ on irregular domains or those localized in an $\varepsilon$-neighborhood of a $d_{\mathcal{M}}$-dimensional smooth manifold $\mathcal{M}\subseteq [0,1]^d$ with $d_{\mathcal{M}}\ll d$. Especially, in the case of an essentially low-dimensional domain, we show an approximation rate $\mathcal{O}\big(\omega_f(\tfrac{\varepsilon}{1-\delta}\sqrt{\tfrac{d}{d_\delta}}+\varepsilon)+\sqrt{d}\,\omega_f(\tfrac{\sqrt{d}}{(1-\delta)\sqrt{d_\delta}}N^{-2/d_\delta}L^{-2/d_\delta})\big)$ for ReLU FNNs to approximate $f$ in the $\varepsilon$-neighborhood, where $d_\delta=\mathcal{O}\big(d_{\mathcal{M}}\tfrac{\ln (d/\delta)}{\delta^2}\big)$ for any $\delta\in(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_{\delta}$-dimensional domain.

翻译：本文以神经元数量来描述深度种子神经网络(FNN) 的近似功率 {DN} 以神经元数量 {FN} 的近似功率 。 通过构建显示,以美元(mathcal) {O\\\\\\\\\\\\\\\\\\\\\\ 底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底