In the process of conducting various physical experiments, a certain set of data is accumulated. Processing and interpreting the simulation results is a fundamental task for analyzing the behaviour of the model, predicting its future actions, and managing the entire system. This paper provides an overview of the currently existing approaches to identification of dynamic systems models: white, gray and black boxes. Special attention is paid to methods based on neural networks. The article suggests a combined approach that allows both preserving a physical consistency of the model and using modern methods for learning from data. A polynomial neural network of a special architecture, approximating the general solution of the system of ordinary differential equations (ODEs) in the form of Taylor map is considered. This model can work in the case of a small amount of initial data, which is a problem when exploiting traditional machine learning methods, and neural networks in particular. The paper presents a new learning approach for PNN based on two steps: reconstructing an ODEs system based on a single trajectory, and identifying a general solution to initialize the weights of a neural network. Neural network representation allows using traditional learning algorithms for additional fine-turning the weights in line with new measured data. A toy example of a nonlinear deflector demonstrates the power and generalization ability of the proposed algorithm.