  Uncertainty propagation in nonlinear dynamic systems remains an outstanding
problem in scientific computing and control. Numerous approaches have been
developed, but are limited in their capability to tackle problems with more
than a few uncertain variables or require large amounts of simulation data. In
this paper, we propose a data-driven method for approximating joint probability
density functions (PDFs) of nonlinear dynamic systems with initial condition
and parameter uncertainty. Our approach leverages on the power of deep learning
to deal with high-dimensional inputs, but we overcome the need for huge
quantities of training data by encoding PDF evolution equations directly into
the optimization problem. We demonstrate the potential of the proposed method
by applying it to evaluate the robustness of a feedback controller for a
six-dimensional rigid body with parameter uncertainty.
