  In many real-world planning problems with factored, mixed discrete and
continuous state and action spaces such as Reservoir Control, Heating
Ventilation, and Air Conditioning, and Navigation domains, it is difficult to
obtain a model of the complex nonlinear dynamics that govern state evolution.
However, the ubiquity of modern sensors allows us to collect large quantities
of data from each of these complex systems and build accurate, nonlinear deep
neural network models of their state transitions. But there remains one major
problem for the task of control -- how can we plan with deep network learned
transition models without resorting to Monte Carlo Tree Search and other
black-box transition model techniques that ignore model structure and do not
easily extend to mixed discrete and continuous domains? In this paper, we
introduce two types of nonlinear planning methods that can leverage deep neural
network learned transition models: Hybrid Deep MILP Planner (HD-MILP-Plan) and
Tensorflow Planner (TF-Plan). In HD-MILP-Plan, we make the critical observation
that the Rectified Linear Unit transfer function for deep networks not only
allows faster convergence of model learning, but also permits a direct
compilation of the deep network transition model to a Mixed-Integer Linear
Program encoding. Further, we identify deep network specific optimizations for
HD-MILP-Plan that improve performance over a base encoding and show that we can
plan optimally with respect to the learned deep networks. In TF-Plan, we take
advantage of the efficiency of auto-differentiation tools and GPU-based
computation where we encode a subclass of purely continuous planning problems
as Recurrent Neural Networks and directly optimize the actions through
backpropagation. We compare both planners and show that TF-Plan is able to
approximate the optimal plans found by HD-MILP-Plan in less computation time...
