We investigate the use of two-layer networks with the rectified power unit, which is called the \(\text {ReLU}^k\) activation function, for function and derivative approximation. By extending and calibrating the corresponding Barron space, we show that two-layer networks with the \(\text {ReLU}^k\) activation function are well-designed to simultaneously approximate an unknown function and its derivatives. When the measurement is noisy, we propose a Tikhonov type regularization method, and provide error bounds when the regularization parameter is chosen appropriately. Several numerical examples support the efficiency of the proposed approach.

我们研究了带有整流功率单元的两层网络的使用，该单元被称为\(\text {ReLU}^k\)激活函数，用于函数和导数逼近。通过扩展和校准相应的巴伦空间，我们表明具有\(\text {ReLU}^k\)激活函数的两层网络经过精心设计，可以同时逼近未知函数及其导数。当测量有噪声时，我们提出了一种 Tikhonov 型正则化方法，并在适当选择正则化参数时提供误差界限。几个数值例子支持所提出方法的效率。