We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function. These bounds provide explicit estimates on the approximation error with respect to the size of the neural networks. We show that tanh neural networks with only two hidden layers suffice to approximate functions at comparable or better rates than much deeper ReLU neural networks.

中文翻译：

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用tanh神经网络逼近函数

我们在高阶 Sobolev 范数中推导出误差的界限，在 Sobolev-regular 的近似以及具有双曲正切激活函数的神经网络的解析函数中产生。这些界限提供了关于神经网络大小的近似误差的明确估计。我们表明，与更深的 ReLU 神经网络相比，只有两个隐藏层的 tanh 神经网络足以以可比或更好的速率逼近函数。