# Learning Dynamics Models with Stable Invariant Sets

arXiv: https://arxiv.org/abs/2006.08935 · HTML: https://mlsanity.com/paper/2006.08935/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2020.

## Built on by (most cited first, 18 of 18)

- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
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- [VeRecycle: Reclaiming Guarantees from Probabilistic Certificates for Stochastic Dynamical Systems after Change](https://mlsanity.com/paper/2505.14001/) (2025)
- [Lyapunov Stable Graph Neural Flow](https://mlsanity.com/paper/2603.12557/) (2026)
- [Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems](https://mlsanity.com/paper/2606.14079/) (2026)
- [Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria](https://mlsanity.com/paper/2610.01356/) (2026)

## Abstract (arXiv)

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.