  Predicting outcomes and planning interactions with the physical world are
long-standing goals for machine learning. A variety of such tasks involves
continuous physical systems, which can be described by partial differential
equations (PDEs) with many degrees of freedom. Existing methods that aim to
control the dynamics of such systems are typically limited to relatively short
time frames or a small number of interaction parameters. We present a novel
hierarchical predictor-corrector scheme which enables neural networks to learn
to understand and control complex nonlinear physical systems over long time
frames. We propose to split the problem into two distinct tasks: planning and
control. To this end, we introduce a predictor network that plans optimal
trajectories and a control network that infers the corresponding control
parameters. Both stages are trained end-to-end using a differentiable PDE
solver. We demonstrate that our method successfully develops an understanding
of complex physical systems and learns to control them for tasks involving PDEs
such as the incompressible Navier-Stokes equations.
