We present a neural network collision checking heuristic, ClearanceNet, and a planning algorithm, CN-RRT. ClearanceNet learns to predict separation distance (minimum distance between robot and workspace) with respect to a workspace. CN-RRT then efficiently computes a motion plan by leveraging three key features of ClearanceNet. First, CN-RRT explores the space by expanding multiple nodes at the same time, processing batches of thousands of collision checks. Second, CN-RRT adaptively relaxes its clearance requirements for more difficult problems. Third, to repair errors, CN-RRT shifts its nodes in the direction of ClearanceNet’s gradient and repairs any residual errors with a traditional RRT, thus maintaining theoretical probabilistic completeness guarantees. In configuration spaces with up to 30 degrees of freedom, ClearanceNet achieves 845x speedup over traditional collision detection methods, while CN-RRT accelerates motion planning by up to 42% over a baseline and finds paths up to 36% more efficient. Experiments on an 11 degree of freedom robot in a cluttered environment confirm the method’s feasibility on real robots.