  We consider neural networks with rational activation functions. The choice of
the nonlinear activation function in deep learning architectures is crucial and
heavily impacts the performance of a neural network. We establish optimal
bounds in terms of network complexity and prove that rational neural networks
approximate smooth functions more efficiently than ReLU networks with
exponentially smaller depth. The flexibility and smoothness of rational
activation functions make them an attractive alternative to ReLU, as we
demonstrate with numerical experiments.
