  Uncertainty is prevalent in robotics. Due to measurement noise and complex
dynamics, we cannot estimate the exact system and environment state. Since
conservative motion planners are not guaranteed to find a safe control strategy
in a crowded, uncertain environment, we propose a density-based method. Our
approach uses a neural network and the Liouville equation to learn the density
evolution for a system with an uncertain initial state. We can plan for
feasible and probably safe trajectories by applying a gradient-based
optimization procedure to minimize the collision risk. We conduct motion
planning experiments on simulated environments and environments generated from
real-world data and outperform baseline methods such as model predictive
control and nonlinear programming. While our method requires offline planning,
the online run time is 100 times smaller compared to model predictive control.
