Proposed are the new types of fast training, scalable analog and digital artificial neural networks (p-networks) based on the new model of formal neuron. The p-network includes synapses with a plurality of weights, and devices of weight selection based on the intensity of the incoming signal. Versions of the p-
networks are presented that are formed with resistance elements, such as, memristor elements. Also described are the matrix methods of training and operation for the proposed network. Training time for the new network is linearly dependent on the size of the network and the volume of data, in contrast to other models of artificial neural networks with the exponential dependence. Thus, p-network training time is dozens time faster than training time of the known networks. The obtained results can be applied in existing artificial neural networks, and in development of a neural microchip.