Swept volume, the volume displaced by a moving object, is an ideal distance metric for sampling-based motion planning because it directly correlates to the amount of motion between two states. However, even approximate algorithms are computationally prohibitive. Our fundamental approach is the application of deep learning to efficiently estimate swept volume computation within a 5%%-10%% error for all robots tested, from rigid bodies to manipulators. However, even inference via the trained network can be computationally costly given the often hundreds of thousands of computations required by sampling-based motion planning. To address this, we demonstrate an efficient hierarchical approach for applying our trained estimator. This approach first pre-filters samples using a weighted Euclidean estimator trained via swept volume. Then, it selectively applies the deep neural network estimator. The first estimator, although less accurate, has metric space properties. The second estimator is a high-fidelity unbiased estimator without metric space properties. We integrate the hierarchical selection approach in both roadmap-based and a tree-based sampling motion planners. Empirical evaluation on the robot set demonstrates that hierarchal application of the metrics yields up to 5000 times faster planning than state of the art swept volume approximation and up to five times higher probability of finding a collision-free trajectory under a fixed time budget than the traditional Euclidean metric.