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Algorithms for motion planning in unknown environments are generally limited in their ability to reason about the structure of the unobserved environment. As such, current methods generally navigate unknown environments by relying on heuristic methods to choose intermediate objectives along frontiers. We present a unified method that combines map prediction and motion planning for safe, time-efficient autonomous navigation of unknown environments by dynamically-constrained robots. We propose a data-driven method for predicting the map of the unobserved environment, using the robot's observations of its surroundings as context. These map predictions are then used to plan trajectories from the robot's position to the goal without requiring frontier selection. We demonstrate that our map-predictive motion planning strategy yields a substantial improvement in trajectory time over a naive frontier pursuit method and demonstrates similar performance to methods using more sophisticated frontier selection heuristics with significantly shorter computation time.