On fixed points of continuous mappings associated with construction of artificial neural networks

A general topological approach is proposed for the construction of converging artificial neural networks (ANN) by applying decision-making algorithms tuned on a sequence of iterations of continuous mappings (ANN layers). The mappings are selected using optimization principles underlying ANN training, and decision-making based on the results of training a multilayer ANN corresponds to finding a sequence converging to a fixed point. It is found that problems of this class are computationally unstable, which is caused by the phenomenon of dynamic chaos associated with the ill-posedness of the problems. Stabilization methods converging to stable fixed points of the mappings are proposed, which is the starting point for a wide variety of mathematical studies concerning the optimization of training sets in ANN construction.