The problem of construction of deep neural networks with the use of genetic algorithms is considered. The problem of structural-parametric synthesis with creation of neural networks is defined. The main purpose of the study is to find a deep neural network that is optimal for solving urgent problems. The classification problem is chosen as the urgent problem to solve. Also the classification of genetic algorithms is given, which are used as a basis for establishing the parameters of deep neural networks A system for the optimal tuning of the parameters of deep neural networks is proposed, which includes a two-stage algorithm. At the first stage of the algorithm, a multicriteria genetic algorithm is selected from a set of possible ones (genetic algorithm of vector estimation, genetic algorithm of Fonseca and Fleming, genetic algorithm of Pareto approximation with niche, genetic algorithm of sorting without dominance, genetic algorithm of Pareto force, genetic algorithm of Pareto-2 force ) that best fits the given training sample. At the second step, the problem of structural-parametric synthesis of a neural network is solved according to the criteria of accuracy and complexity. As a result of training, the values of the neural network parameters are found, such as: the number of layers, the number of neurons in each layer, the values of the weight coefficients.The modeling of the proposed system is carried out. The results of modeling, comparison of results with similar software packages are presented. The obtained results show the possibilities of wide use.