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

TL;DR

The paper introduces a new neural network architecture based on port-Hamiltonian systems that can model multiple stable equilibria by using a product of Bregman divergences from an input-convex network. This design enables the representation of complex energy landscapes with multiple attractors, such as double-well potentials, while maintaining local Lyapunov stability. The method improves convergence speed and accurately recovers energy surface characteristics in benchmark systems.

Why It Matters

Many real-world systems exhibit multiple stable states, but traditional stable neural models are limited to single attractors. This advancement allows for more accurate and physically consistent modeling of such systems, improving both stability guarantees and learning efficiency in control and dynamical systems applications.

Key Findings

• The proposed model uses a product of Bregman divergences to parametrize the Hamiltonian, enabling representation of systems with multiple asymptotically stable equilibria.
• The system is locally Lyapunov stable, and the presence of multiple stable equilibria necessitates additional unstable equilibria, which are mathematically constrained.
• All equilibria are bounded, and under hyperbolicity assumptions, almost-everywhere stability is achieved, with empirical results showing up to 8.5× faster convergence.