In this article, two novel deep learning methods are proposed for displacement estimation in ultrasound elastography (USE). Although convolutional neural networks (CNNs) have been very successful for displacement estimation in computer vision, they have been rarely used for USE. One of the main limitations is that the radio frequency (RF) ultrasound data, which is crucial for precise displacement estimation, has vastly different frequency characteristics compared with images in computer vision. Top-rank CNN methods used in computer vision applications are mostly based on a multilevel strategy, which estimates finer resolution based on coarser ones. This strategy does not work well for RF data due to its large high-frequency content. To mitigate the problem, we propose modified pyramid warping and cost volume network (MPWC-Net) and RFMPWC-Net, both based on PWC-Net, to exploit information in RF data by employing two different strategies. We obtained promising results using networks trained only on computer vision images. In the next step, we constructed a large ultrasound simulation database and proposed a new loss function to fine-tune the network to improve its performance. The proposed networks and well-known optical flow networks as well as state-of-the-art elastography methods are evaluated using simulation, phantom, and in vivo data. Our two proposed networks substantially outperform current deep learning methods in terms of contrast-to-noise ratio (CNR) and strain ratio (SR). Also, the proposed methods perform similar to the state-of-the-art elastography methods in terms of CNR and have better SR by substantially reducing the underestimation bias.