In this work, we investigate a mathematical model to reconstruct the mechanical properties of a heterogeneous elastic medium for the optical coherence elastography imaging modality. To this end, we propose machine learning tools by exploring neural networks to solve the inverse problem of elastography. Our algorithm updates the parameters combining the backpropagation technique with the ADAM optimizer to minimize a cost function which is defined using a fully discretized scheme of the direct problem. In our framework, we analyze the theoretical relative error between the exact solution and the numerical approximation given by the respective algorithm for the case of noise-free data and noisy data. We report several computational results using fabricated data with and without noise.