We present a data-driven approach to compensate for optical aberrations in calibration-free quantitative phase imaging (QPI). Unlike existing methods that require additional measurements or a background region to correct aberrations, we exploit deep learning techniques to model the physics of aberration in an imaging system. We demonstrate the generation of a single-shot aberration-corrected field image by using a U-net-based deep neural network that learns a translation between an optical field with aberrations and an aberration-corrected field. The high fidelity and stability of our method is demonstrated on 2D and 3D QPI measurements of various confluent eukaryotic cells and microbeads, benchmarking against the conventional method using background subtractions.