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

Simon Heilig; Jens Püttschneider; Mohammad Itani; Asja Fischer; Timm Faulwasser

cs.LGOct 1, 2026

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

Authors: Simon Heilig, Jens Püttschneider, Mohammad Itani, Asja Fischer, Timm Faulwasser

Organizations: Faculty of Computer Science, Ruhr University Bochum, Germany · Institute of Control Systems, Hamburg University of Technology, Germany

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 $\times$ .

Figures & tables

Panel of Figure 1 — Figure 1 : Energy landscape and two trajectories of the Duffing oscillator for (a) the fitted s-PHNN and (b) our ms-PHNN, showing that the ms-PHNN captures both asymptotically stable equilibria, whereas the s-PHNN learns only one. (c) Training and validation derivative loss for neural ODE (NODE), s-PHNN, and our ms-PHNN, median over 10 seeds.

System	Model	train	validation	test	speedup
Duffing	NODE	$0.0033{\scriptstyle\,\pm\,0.0009}$	$0.0049{\scriptstyle\,\pm\,0.0009}$	$0.0039{\scriptstyle\,\pm\,0.0008}$	$1\times$
	s-PHNN	$0.0850{\scriptstyle\,\pm\,0.0165}$	$0.0888{\scriptstyle\,\pm\,0.0179}$	$0.0883{\scriptstyle\,\pm\,0.0187}$	—
	ms-PHNN	$\mathbf{0.0008}{\scriptstyle\,\pm\,0.0007}$	$\mathbf{0.0010}{\scriptstyle\,\pm\,0.0007}$	$\mathbf{0.0009}{\scriptstyle\,\pm\,0.0007}$	${8.5}{\scriptstyle\,\pm\,2.1}$
Asym. Duffing	NODE	$0.0050{\scriptstyle\,\pm\,0.0022}$	$0.0084{\scriptstyle\,\pm\,0.0025}$	$0.0065{\scriptstyle\,\pm\,0.0028}$	$1\times$
	s-PHNN	$0.0607{\scriptstyle\,\pm\,0.0198}$	$0.0649{\scriptstyle\,\pm\,0.0232}$	$0.0640{\scriptstyle\,\pm\,0.0217}$	—
	ms-PHNN	$\mathbf{0.0031}{\scriptstyle\,\pm\,0.0015}$	$\mathbf{0.0041}{\scriptstyle\,\pm\,0.0020}$	$\mathbf{0.0034}{\scriptstyle\,\pm\,0.0017}$	$2.7{\scriptstyle\,\pm\,0.6}$

Table 1 : Derivative RMSE as mean

\pm

std over 10 seeds, and the optimiser-step speedup over NODE final loss. Bold marks the best model. "—" means the model did not reach the reference loss.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

System	min	saddle	max	$\lambda_{\min}(\nabla^{2}\mathcal{H})$ at min	$\lambda_{\min}(\nabla^{2}\mathcal{H})$ other	$\min_{i}\lvert\operatorname{Re}\lambda_{i}(\bm{A})\rvert$
	located / true
Duffing	$2/2$	$1/1$	$0/0$	$[0.028,\,0.073]$	$[-0.073,\,-0.028]$	$\geq 0.471$
Asym. Duffing	$2/2$	$1/1$	$0/0$	$[0.005,\,0.059]$	$[-0.069,\,-0.006]$	$\geq 0.320$
Four-Magnets	$4/4$	$4/4$	$1/1$	$[0.504,\,0.902]$	$[-2.330,\,-0.703]$	$\geq 0.417$

Appendix

Table 2 : Numerical verification of Assumption 2 over

10

seeds per system inspecting the eigenspectrum of critical points obtained by a fitted ms-PHNN.

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Quantum Port-Hamiltonian Neural Networks: Learning Conservative and Dissipative Dynamics via Measurement-Induced Nonlinearity

Dibakar Sigdel

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Quantum Neural Networks Hamiltonian Quantum Circuits Nonlinearity Chaos System Identification

Open paper AskPDF arXiv

We introduce Quantum Port-Hamiltonian Neural Networks (Q-pHNNs), a family of parameterised quantum circuits that learn classical dynamics in a structure-preserving manner. The framework relies on the Isomorphic Hamiltonian Mapping (IHM): the skew-symmetric interconnection matrix

\mathbf{J}

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\mathbf{R}

corresponds to Measurement-Induced NonLinearity (MINL) realised via mid-circuit measurement and classical feedforward. This ensures conservation and passivity are enforced by construction rather than penalty terms. We instantiate the IHM in four architectures: (1) a Quantum HNN that learns conservative energy manifolds and extracts Hamilton's equations exactly via the Parameter-Shift Rule; (2) a Q-pHNN using Born-rule measurement for dissipation; (3) a Q-pHNN jointly learning the energy ansatz and damping coefficient; and (4) a topology-entangled Quantum Graph Neural Network for

N

-node coupled-phasor networks. Experiments on the nonlinear pendulum and damped harmonic oscillator demonstrate: (i) ~~$1.35\%$ relative energy drift with a symplectic integrator and scale correction; (ii)~~

100\%

energy monotonicity for the MINL circuit; and (iii)~

12.1\%

error in damping-coefficient identification from vector-field snapshots with no direct supervision on the damping coefficient.

Abstract

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