两层神经网络可精准逼近任意连续函数，且有几何构造方法。
Geometric separation and constructive universal approximation with two hidden layers
- 用几何方法构造两层网络实现集合分离
- 两层网络可任意精度逼近任意紧集上的连续函数
- 对有限点集有更简洁的单隐层结果，适合理论研究
我们给出了一种神经网络的几何构造方法，用于分离ℝⁿ中不相交的紧集，并据此获得一个可构造的通用逼近定理。具体而言，我们证明了具有两层隐藏层、采用S型激活函数（即严格单调的有界连续函数）或ReLU激活函数的网络，能够以任意精度在一致范数下逼近ℝⁿ中任意紧集K上的任意实值连续函数。当K为有限集时，构造可简化，得到一个精确的深度为2（单隐藏层）的逼近结果。
原文摘要 · Abstract (English)
We give a geometric construction of neural networks that separate disjoint compact subsets of $\Bbb R^n$, and use it to obtain a constructive universal approximation theorem. Specifically, we show that networks with two hidden layers and either a sigmoidal activation (i.e., strictly monotone bounded continuous) or the ReLU activation can approximate any real-valued continuous function on an arbitrary compact set $K\subset\Bbb R^n$ to any prescribed accuracy in the uniform norm. For finite $K$, the construction simplifies and yields a sharp depth-2 (single hidden layer) approximation result.
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