Shear wave elastography involves applying a non-invasive acoustic radiation force to the tissue and imaging the induced deformation to infer its mechanical properties. This work investigates the use of convolutional neural networks to improve displacement estimation accuracy in shear wave imaging. Our training approach is completely unsupervised, which allows to learn the estimation of the induced micro-scale deformations without ground truth labels. We also present an ultrasound simulation dataset where the shear wave propagation has been simulated via finite element method. Our dataset is made publicly available along with this paper, and consists in 150 shear wave propagation simulations in both homogenous and hetegeneous media, which represents a total of 20,000 ultrasound images. We assessed the ability of our learning-based approach to characterise tissue elastic properties (i.e., Young's modulus) on our dataset and compared our results with a classical normalised cross-correlation approach.