We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval $(-1,1)$. We assume that the given source term and reaction coefficient are analytic in $[-1,1]$. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation parameter for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and $\tanh$- and sigmoid-activated NNs. The latter activations can represent ``exponential boundary layer solution features'' explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. We prove that all DNN architectures allow robust exponential solution expression in so-called `energy' as well as in `balanced' Sobolev norms, for analytic input data.

翻译：我们证明了在索伯列夫范数下，有界区间$(-1,1)$上一类奇异摄动椭圆型两点边值问题解集的深度神经网络（简称DNN）表达能力速率界。假设给定源项和反应系数在$[-1,1]$上解析，针对若干类DNN架构，建立了关于神经网络规模且在奇异摄动参数上一致成立的索伯列夫范数表达能力速率界。具体包括ReLU神经网络、脉冲神经网络、以及双曲正切和S型激活函数神经网络。后两种激活函数能够在DNN最后一个隐藏层（即浅层子网络）中显式表示"指数边界层解特征"，并在神经网络规模上实现了更优的鲁棒表达能力速率界。我们证明，对于解析输入数据，所有DNN架构都能在所谓的"能量"范数和"平衡"索伯列夫范数下实现鲁棒的指数解表达。