Phase unwrapping is a critical step of interferometric synthetic aperture radar processing, and its accuracy directly determines the reliability of subsequent applications. Many phase unwrapping methods have been proposed, most of which assume that the phase has spatial continuity, while decorrelation noise and aliasing fringes invalidate the assumptions, resulting in poor performance of these methods. To obtain more reliable unwrapping results, in this article, a deep convolutional neural network, called a discontinuity estimation network (DENet), is proposed for predicting the probabilities of phase discontinuities in interferograms. The main advantages of DENet are: 1) using branching structure to extract detailed and high-level features separately and retain details while making full use of contextual information; 2) using multichannel input, including interferogram, range/azimuthal phase gradients, and residues map, to provide effective guidance for discontinuity prediction; and 3) using a single network to estimate phase discontinuities in both range and azimuth directions simultaneously. To train the network, a dataset simulation strategy is proposed to generate enough training samples. The strategy considers a variety of phase components, such as terrain-related phase, random deformation, atmospheric turbulence, and noise. The phase discontinuity estimated by DENet is then converted to costs in the minimum cost flow (MCF) solver of the statistical-cost, network-flow algorithm for phase unwrapping (SNAPHU) to obtain the final unwrapped phase. Based on validations of simulated and real interferograms, the proposed method exhibits excellent performance compared to traditional and deep learning unwrapping methods. The proposed method can effectively unwrap large-scale, low-quality interferograms, which is expected to significantly improve the accuracy of synthetic aperture radar interferometry (InSAR) applications.