The object of the study are neural networks. The goal is to understand and develop a recognition system of mathematical formulas and symbols. The article deals with the basic provisions of neural networks, which principle of work is based on functioning neural networks of the human brain. We describe a mathematical representation of the pulse passing through the synaptic connections of neurons and their processing for further action. It describes problems at creation and use of neural networks, which lead to restriction of the scope of information being processed, complication of the structure and the emergence of new tasks to be solved. The article also suggests some ways to resolve these difficulties. Based on the study, the authors review a stepwise algorithm of the neural network, which allows recognizing the symbols and mathematical expressions. The steps of the algorithm are disassembled using specific examples to demonstrate the principle of its operation.