The paper utilizes a nonlinear viscoelastic model of the medium with associative and hereditary memory in the form of a system of integro-differential equations. The hereditary memory is contained (for a long time) in the Volterra integral operator and the associative (short-term) memory is determined by the differential operator. Identification of the model is solved using neural networks in the version of the finite-dimensional approximation with discrete time for a composite material based on a matrix of natural rubber (polyisoprene), filled by 20% with N-330 carbon black. The study is carried out both in the small strain mode and in the finite strain mode. The issues studied in the paper are the accuracy of reproduction by the model of the real nonlinearity function and its ability to generalize the experimental material based on the training sample. The identification of this model showed a good reproduction of the actual function of nonlinearity of a real viscoelastic material in the finite strain mode.