  In this paper, we analyze the number of neurons and training parameters that
a neural networks needs to approximate multivariate functions of bounded second
mixed derivatives -- Korobov functions. We prove upper bounds on these
quantities for shallow and deep neural networks, breaking the curse of
dimensionality. Our bounds hold for general activation functions, including
ReLU. We further prove that these bounds nearly match the minimal number of
parameters any continuous function approximator needs to approximate Korobov
functions, showing that neural networks are near-optimal function
approximators.
