Learning of Neural Network with Optimal Control Tools

Any learned artificial neural network on a given set of observations represents a function of several variables with vector values or real values. However, in general it is unknown except very simply cases and we have trouble to tell anything about its properties behind very general results received from learning data. In applications, such as medicine, it needs to say not only that on training data we get some error, we have to know that an error is not greater than some ɛ for all data for which we consider the system.