How does uncertainty propagate in dynamical systems? We extend adjoint-based a posteriori analysis for differential operators of generic dynamical systems acting on states to the Liouville operator acting on probability densities of the states resulting in theoretically rigorous estimates of sensitivity and error for a broad class of computed QoI in propagating uncertainty through dynamical systems. To make these estimates computationally tractable, normaliz- ing flows exploiting the hyperbolic nature of the Liouville equation are utilized to construct a spatio-temporal density estimator via a loss function defined using the Kullback-Leibler (KL) divergence and a physics informed residual loss. We demonstrate our methods on 3 dynamical systems – linear and non-linear two dimensional system and a reduced order model, on multiple quantities of interest