  Deep networks are commonly used to model dynamical systems, predicting how
the state of a system will evolve over time (either autonomously or in response
to control inputs). Despite the predictive power of these systems, it has been
difficult to make formal claims about the basic properties of the learned
systems. In this paper, we propose an approach for learning dynamical systems
that are guaranteed to be stable over the entire state space. The approach
works by jointly learning a dynamics model and Lyapunov function that
guarantees non-expansiveness of the dynamics under the learned Lyapunov
function. We show that such learning systems are able to model simple dynamical
systems and can be combined with additional deep generative models to learn
complex dynamics, such as video textures, in a fully end-to-end fashion.
