Hopfield neural networks as port-Hamiltonian and gradient systems

Arjan van der Schaft

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arXiv · 2601.02951

Hopfield neural networks as port-Hamiltonian and gradient systems

Arjan van der Schaft

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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.

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Arjan van der Schaft. 2026-01-06. Hopfield neural networks as port-Hamiltonian and gradient systems. https://arxiv.org/abs/2601.02951

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