The essential order of simultaneous approximation for neural networks

By establishing both upper and lower bound estimations on simultaneous approximation order, the essential simultaneous approximation order of the network which can simultaneously approximate function and its derivatives is estimated and the theorem of saturation (the largest capacity of simultaneous approximation) is proved. These results can precisely characterize the simultaneous approximation ability of the network and the relationship among the rate of simultaneous approximation, the topological structure of hidden layer and the properties of approximated function.