# Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria

arXiv: https://arxiv.org/abs/2610.01356 · HTML: https://mlsanity.com/paper/2610.01356/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 22 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 13 of 13)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [The Lyapunov Neural Network: Adaptive Stability Certification for Safe Learning of Dynamical Systems](https://mlsanity.com/paper/1808.00924/) (2018)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Learning Dynamics Models with Stable Invariant Sets](https://mlsanity.com/paper/2006.08935/) (2020)
- [Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2212.00893/) (2022)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Safe Physics-Informed Machine Learning for Dynamics and Control](https://mlsanity.com/paper/2504.12952/) (2025)
- [Nonlinear port-Hamiltonian system identification from input-state-output data (ISO-pHNN)](https://mlsanity.com/paper/2501.06118/) (2025)
- [Solving the Dirichlet problem for the Monge-Amp`ere equation using neural networks](https://mlsanity.com/paper/2110.03310/) (2021)

## Abstract (arXiv)

Stable port-Hamiltonian neural networks certify asymptotic stability by construction. Yet, their Hamiltonian is a global Lyapunov function with a single global minimum, so they can represent only dynamic systems with one attractor. We demonstrate that this excludes even simple systems with energy landscapes forming a double well, and we overcome the restriction by parametrising the Hamiltonian as a product of Bregman divergences generated by one input-convex network. We prove that the resulting model is locally Lyapunov stable, that the coexistence of stable equilibria forces additional non-asymptotically-stable equilibria to exist, that all equilibria lie in a bounded region, and under a hyperbolicity assumption that almost-everywhere stability holds. On three systems our approach is able to recover the energy surface characteristics and improve the convergence speed by 1.8×-8.5×.