The paper considers several approaches to training multilayer fully connected neural networks. In particular, the authors have developed an artificial neural network, the purpose of which is to recognize images of numbers from zero to six. To train the neural network, a training set was formed. The authors carried out a number of experiments on the implementation of various learning methods for the considered artificial neural network. The description of the network training procedure using the classical genetic algorithm is given. The results showed that the genetic algorithm in the classical form is ineffective for solving the problem, since the training time of the artificial neural network is significantly higher compared to the backpropagation algorithm. A combined learning method based on a genetic algorithm and gradient descent has also been proposed. The results of the experiment showed close results in terms of efficiency in comparison with the backpropagation algorithm. It follows from this that the genetic algorithm is applicable for solving the problems of training artificial neural networks.