This paper is concerned with the application of a machine learning approach to inverse elastic scattering problems via neural networks. In the forward problem, the displacements are approximated by linear combinations of the fundamental tensors of the Cauchy-Navier equations of elasticity, which are expressed in terms of sources placed inside the elastic solid. From the near-field measurement data, a two-layer neural network method consisting of a gated recurrent unit to gate recurrent unit has been used to reconstruct the shape of an unknown elastic body. Moreover, the convergence of the method is proved. Finally, the feasibility and effectiveness of the presented method are examined through numerical examples.