We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper bounds for the complexity of the function class of derivatives of RePUs networks. We establish error bounds for simultaneously approximating $C^s$ smooth functions and their derivatives using RePU-activated deep neural networks. Furthermore, we derive improved approximation error bounds when data has an approximate low-dimensional support, demonstrating the ability of RePU networks to mitigate the curse of dimensionality. To illustrate the usefulness of our results, we consider a deep score matching estimator (DSME) and propose a penalized deep isotonic regression (PDIR) using RePU networks. We establish non-asymptotic excess risk bounds for DSME and PDIR under the assumption that the target functions belong to a class of $C^s$ smooth functions. We also show that PDIR achieves the minimax optimal convergence rate and has a robustness property in the sense it is consistent with vanishing penalty parameters even when the monotonicity assumption is not satisfied. Furthermore, if the data distribution is supported on an approximate low-dimensional manifold, we show that DSME and PDIR can mitigate the curse of dimensionality.

翻译：研究整流幂函数（RePU）激活的可微分神经网络性质。我们证明RePU神经网络偏导数可由混合RePU激活网络表示，并给出RePU网络导数函数类复杂度的上界。建立利用RePU激活深度神经网络同时逼近$C^s$光滑函数及其导数的误差界。进一步，当数据具有近似低维支撑时，推导改进的逼近误差界，证明RePU网络能缓解维度灾难。为阐明结果实用性，提出深度分数匹配估计量（DSME）并设计基于RePU网络的惩罚深度等渗回归（PDIR）。在目标函数属于$C^s$光滑函数类的假设下，建立DSME与PDIR的非渐近超额风险界。证明PDIR达到极小化最优收敛速度，且当单调性假设不满足时仍具有鲁棒性——随惩罚参数趋零而保持一致性。进一步，若数据分布支撑于近似低维流形，则DSME与PDIR可缓解维度灾难。