Diagnostics of objects and situations, the choice of solutions - concepts and challenges in the modeling of decision rules of recognition, factor analysis and optimization of parameters of these models. This paper examines conceptual issues of the theory of artificial neural networks, including their correctness and stability. Correctness - ensuring required reactions or small variations of the network or system to input information on a material of training. Sustainability is the preservation of the reactions of a neural network or a small change at small changes of the initial information. Will only consider layered networks - they use superposition functions. Probabilistic methods in this article are not used.