  Invariance and stability are essential notions in dynamical systems study,
and thus it is of great interest to learn a dynamics model with a stable
invariant set. However, existing methods can only handle the stability of an
equilibrium. In this paper, we propose a method to ensure that a dynamics model
has a stable invariant set of general classes such as limit cycles and line
attractors. We start with the approach by Manek and Kolter (2019), where they
use a learnable Lyapunov function to make a model stable with regard to an
equilibrium. We generalize it for general sets by introducing projection onto
them. To resolve the difficulty of specifying a to-be stable invariant set
analytically, we propose defining such a set as a primitive shape (e.g.,
sphere) in a latent space and learning the transformation between the original
and latent spaces. It enables us to compute the projection easily, and at the
same time, we can maintain the model's flexibility using various invertible
neural networks for the transformation. We present experimental results that
show the validity of the proposed method and the usefulness for long-term
prediction.
