Learning Dynamics Models with Stable Invariant Sets
This work addresses a limitation in dynamical systems modeling for researchers and practitioners by enabling stable learning of more complex invariant sets, though it is incremental as it builds on prior methods.
The paper tackles the problem of learning dynamics models with stable invariant sets beyond just equilibria, such as limit cycles and line attractors, by generalizing a Lyapunov-based approach with latent space projections and invertible neural networks, achieving valid long-term prediction in experiments.
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.