This paper studies the approximation property of ReLU neural networks (NNs) to piecewise constant functions with unknown interfaces in bounded regions in $\mathbb{R}^d$. Under the assumption that the discontinuity interface $\Gamma$ may be approximated by a connected series of hyperplanes with a prescribed accuracy $\varepsilon >0$, we show that a three-layer ReLU NN is sufficient to accurately approximate any piecewise constant function and establish its error bound. Moreover, if the discontinuity interface is convex, an analytical formula of the ReLU NN approximation with exact weights and biases is provided.

翻译：本文研究了ReLU神经网络在有界区域$\mathbb{R}^d$中对具有未知间断面的分段常数函数的逼近性质。在假设间断界面$\Gamma$可由一系列连通超平面以给定精度$\varepsilon >0$逼近的前提下，我们证明三层ReLU神经网络足以精确逼近任意分段常数函数，并建立了其误差界。此外，若间断界面为凸集，本文给出了具有精确权重与偏置的ReLU神经网络逼近解析表达式。