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This paper aims to improve the computational efficiency of motion planning for mobile robots with non-trivial dynamics through the use of learned controllers. Offline, a system-specific controller is first trained in an empty environment. Then, for the target environment, the approach constructs a data structure, a “Roadmap with Gaps,” to approximately learn how to solve planning queries using the learned controller. The roadmap nodes correspond to local regions. Edges correspond to applications of the learned controller that approximately connect these regions. Gaps arise as the controller does not perfectly connect pairs of individual states along edges. Online, given a query, a tree sampling-based motion planner uses the roadmap so that the tree’s expansion is informed towards the goal region. The tree expansion selects local subgoals given a wavefront on the roadmap that guides towards the goal. When the controller cannot reach a subgoal region, the planner resorts to random exploration to maintain probabilistic completeness and asymptotic optimality. The accompanying experimental evaluation shows that the approach significantly improves the computational efficiency of motion planning on various benchmarks, including physics-based vehicular models on uneven and varying friction terrains as well as a quadrotor under air pressure effects.