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

2026-10-01

AI Summary

This study addresses the limitation of conventional port-Hamiltonian neural networks, which can represent only a single attractor and thus fail to model multistable systems. We propose a modeling framework that supports multiple asymptotically stable equilibria by parameterizing the Hamiltonian as a product of Bregman divergences. By integrating input-convex neural networks with port-Hamiltonian theory, our approach overcomes the restriction to a single global minimum. We rigorously establish local Lyapunov stability and almost-everywhere convergence. Experimental results demonstrate that the proposed model successfully recovers the characteristics of multistable energy landscapes, achieving convergence speeds 1.8 to 8.5 times faster than baseline methods.

Abstract

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$\times$-8.5$\times$.

Problem

• Port-Hamiltonian Neural Networks
• Multiple Equilibria
• Asymptotic Stability
• Lyapunov Function
• Double Well Potential

Innovation

• Port-Hamiltonian Neural Networks
• Multiple Asymptotically Stable Equilibria
• Bregman Divergences
• Input-Convex Network
• Lyapunov Stability