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We study the properties of differentiable neural networks activated byrectified power unit (RePU) functions. We show that the partial derivatives ofRePU neural networks can be represented by RePUs mixed-activated networks andderive upper bounds for the complexity of the function class of derivatives ofRePUs networks. We establish error bounds for simultaneously approximating$C^s$ smooth functions and their derivatives using RePU-activated deep neuralnetworks. Furthermore, we derive improved approximation error bounds when datahas an approximate low-dimensional support, demonstrating the ability of RePUnetworks to mitigate the curse of dimensionality. To illustrate the usefulnessof our results, we consider a deep score matching estimator (DSME) and proposea penalized deep isotonic regression (PDIR) using RePU networks. We establishnon-asymptotic excess risk bounds for DSME and PDIR under the assumption thatthe target functions belong to a class of $C^s$ smooth functions. We also showthat PDIR has a robustness property in the sense it is consistent withvanishing penalty parameters even when the monotonicity assumption is notsatisfied. Furthermore, if the data distribution is supported on an approximatelow-dimensional manifold, we show that DSME and PDIR can mitigate the curse ofdimensionality.

Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regression