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The paper considers the neural network approach for solving the Cauchy problem for ordinary differential equations of the first order based on the representation of the function as a superposition of elementary functions, the algorithm of solving the problem is proposed. The application of the neural network approach allows obtaining the desired solution in the form of a functional dependence that satisfies smoothness conditions. On the basis of a two-layer perceptron, a model of a neural network solution of the problem and a numerical algorithm realizing the search for a solution are built. We developed a program and algorithmic solution of the Cauchy problem. We analyzed the accuracy of the results and its interrelation with the parameters of the neural networks used. The equivalence of the work of the neural network algorithm and the third-order numerical Runge-Kutta algorithm is shown. In addition, the problem of retraining the neural network algorithm for solving the Cauchy problem for first order ordinary differential equations is posed.