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

The motivational example is compelling: the authors show that an s-PHNN trained on the Duffing oscillator learns only one of its two stable equilibria because the architecture is globally asymptotically stable by design. The proposed fix---a multiplicative Hamiltonian $H_{\bm{\theta}}(\bm{x})=\prod_{k=1}^{m}D_{\bm{\theta},k}(\bm{x})$ built from a single FICNN---is simple yet effective, and the theory developed around it is rigorous. Local asymptotic stability of the target equilibria is proven via standard Lyapunov arguments, the existence and boundedness of extra equilibria are established via the mountain pass theorem and gradient bounds, and the almost-global convergence claim is defended with the stable manifold theorem. The separable canonical variant for mechanical systems is a useful specialization that relaxes the dissipation requirement to the momenta only.

The almost-global asymptotic stability guarantee in Theorem 2 hinges on Assumption 2, which requires that every non-target equilibrium be hyperbolic with at least one unstable direction. This is a strong, non-constructive prior: the authors verify it only numerically after training for their specific benchmarks, and it is unclear how to enforce it during learning in general.
The experiments are also very small-scale---the state dimensions are 2 and 4---and the model is given the number and approximate locations of the equilibria at initialization. The harder four-magnet task already shows the ms-PHNN trailing the unconstrained NODE in test derivative RMSE, suggesting the inductive bias can become a liability when the landscape is more complex or capacity is limited. The s-PHNN baseline performs so poorly that it functions more as a strawman than a competitive benchmark; comparisons to richer baselines (e.g., ensembles or region-of-attraction methods) are missing.

The evidence clearly supports the central claim that the multiplicative architecture captures multiple stable equilibria where the s-PHNN cannot. Figure 1 and Table 1 show that the ms-PHNN recovers the correct energy surface on the Duffing oscillator and achieves lower derivative RMSE than both s-PHNN and, in two of three cases, the unconstrained NODE. The convergence speedup of $1.8\times$-$8.5\times$ relative to NODE is measured by the number of optimizer steps needed to reach the final NODE loss level on a derivative-fitting objective; this is a valid metric, but it does not establish superiority on long-term trajectory rollouts, which are not reported. The comparison to related work is fair in establishing the architectural limitation of convex-Hamiltonian approaches, yet it omits discussion of whether alternative multi-modal Lyapunov constructions or mixture models could achieve similar guarantees without the product structure.

The authors describe the network architecture, optimizer settings, and data generation in Appendix C. However, no code repository or data link is provided, and several details that matter for reproduction are underspecified: the exact initialization scheme for the FICNN weight matrices, the precise Gaussian noise scale used to perturb the ground-truth equilibria at initialization, and the rejection-sampling code for the batch-covering scheme. Because the stability proofs rely on architectural constraints (non-negative weights in the FICNN, enforced skew-symmetry of $\bm{J}_{\bm{\theta}}$, positive definiteness of $\bm{R}_{\bm{\theta}}$), an independent reimplementation without reference code is error-prone.