We present a method to reconstruct a dense spatiotemporal depth map of a non-rigidly deformable object directly from a video sequence. The estimation of depth is performed locally on spatio-temporal patches of the video, and then the full depth video of the entire shape is recovered by combining them together. Since the geometric complexity of a local spatiotemporal patch of a deforming non-rigid object is often simple enough to be faithfully represented with a parametric model, we artificially generate a database of small deforming rectangular meshes rendered with different material properties and light conditions, along with their corresponding depth videos, and use such data to train a convolutional neural network. We tested our method on both synthetic and Kinect data and experimentally observed that the reconstruction error is significantly lower than the one obtained using conventional non-rigid structure from motion approaches.