  Density of the reachable states can help understand the risk of
safety-critical systems, especially in situations when worst-case reachability
is too conservative. Recent work provides a data-driven approach to compute the
density distribution of autonomous systems' forward reachable states online. In
this paper, we study the use of such approach in combination with model
predictive control for verifiable safe path planning under uncertainties. We
first use the learned density distribution to compute the risk of collision
online. If such risk exceeds the acceptable threshold, our method will plan for
a new path around the previous trajectory, with the risk of collision below the
threshold. Our method is well-suited to handle systems with uncertainties and
complicated dynamics as our data-driven approach does not need an analytical
form of the systems' dynamics and can estimate forward state density with an
arbitrary initial distribution of uncertainties. We design two challenging
scenarios (autonomous driving and hovercraft control) for safe motion planning
in environments with obstacles under system uncertainties. We first show that
our density estimation approach can reach a similar accuracy as the
Monte-Carlo-based method while using only 0.01X training samples. By leveraging
the estimated risk, our algorithm achieves the highest success rate in goal
reaching when enforcing the safety rate above 0.99.
