In this work, we propose a novel missile guidance algorithm that combines deep learning based trajectory prediction with nonlinear model predictive control. Although missile guidance and threat interception is a well-studied problem, existing algorithms' performance degrade significantly when the target is pulling high acceleration attack maneuvers while rapidly changing its direction. We argue that since most threats execute similar attack maneuvers, these nonlinear trajectory patterns can be processed with modern machine learning methods to build high accuracy trajectory prediction algorithms. We train a long short-term memory network (LSTM) based on a class of simulated structured agile attack patterns, then combine this predictor with quadratic programming based nonlinear model predictive control (NMPC). Our method, named nonlinear model based predictive control with target acceleration predictions (NMPC-TAP), significantly outperforms compared approaches in terms of miss distance, for the scenarios where the target/threat is executing agile maneuvers.