Hopfield neural networks as port-Hamiltonian and gradient systems
Authors
Abstract
The structure of continuous Hopfield networks is revisited from a system-theoretic point of view. After adopting a novel electrical network interpretation involving nonlinear capacitors, it is shown that Hopfield networks admit a port-Hamiltonian formulation provided an extra passivity condition is satisfied. Subsequently it is shown that any Hopfield network can be represented as a gradient system, with Riemannian metric given by the inverse of the Hessian matrix of the total energy stored in the nonlinear capacitors. On the other hand, the well-known ‘energy’ function employed by Hopfield turns out to be the dissipation potential of the gradient system, and this potential is shown to satisfy a dissipation inequality that can be used for analysis and interconnection.
Citation
- Journal: IEEE Transactions on Automatic Control
- Year: 2026
- Volume:
- Issue:
- Pages: 1–4
- Publisher: Institute of Electrical and Electronics Engineers (IEEE)
- DOI: 10.1109/tac.2026.3740658
BibTeX
@article{van_der_Schaft_2026,
title={{Hopfield neural networks as port-Hamiltonian and gradient systems}},
ISSN={2334-3303},
DOI={10.1109/tac.2026.3740658},
journal={IEEE Transactions on Automatic Control},
publisher={Institute of Electrical and Electronics Engineers (IEEE)},
author={van der Schaft, Arjan},
year={2026},
pages={1--4}
}