  We introduce a method for learning provably stable deep neural network based
dynamic models from observed data. Specifically, we consider discrete-time
stochastic dynamic models, as they are of particular interest in practical
applications such as estimation and control. However, these aspects exacerbate
the challenge of guaranteeing stability. Our method works by embedding a
Lyapunov neural network into the dynamic model, thereby inherently satisfying
the stability criterion. To this end, we propose two approaches and apply them
in both the deterministic and stochastic settings: one exploits convexity of
the Lyapunov function, while the other enforces stability through an implicit
output layer. We demonstrate the utility of each approach through numerical
examples.
