A recent study shows that port‑Hamiltonian models built from both analytic expression libraries and neural networks can outperform models that use only one representation. The authors compared four fitting approaches: gradient‑based optimisation with an integration‑scheme loss, gradient‑based optimisation with a roll‑out loss, sparse regression with ridge regularisation and term removal, and sparse regression with a midpoint fit.

For a two‑mass system with constant damping and an unknown input map, the hybrid analytic‑neural network model trained with the gradient‑based integration‑scheme loss achieved an NRMSE of 6.01 × 10⁻⁵ on noise‑free data, while a purely analytic model trained with the same method scored 4.02 × 10⁻². The neural‑network model alone improved from 1.10 × 10⁻² to 1.72 × 10⁻² when the data were smoothed and the roll‑out loss was used.

In a related system with state‑dependent damping and a known input map, the analytic gradient‑based scheme yielded a mean NRMSE of 1.51 × 10⁻³, whereas the hybrid analytic‑neural network model reached 3.88 × 10⁻⁴.

However, analytic models fitted by gradient‑based optimisation failed to extrapolate beyond t = 20 seconds in some tests, so the authors report prediction errors only up to that time. This limitation arises because no global lower bound on the Hamiltonian was imposed for purely analytic models, which can compromise passivity and boundedness of trajectories. In contrast, neural‑network Hamiltonian models are bounded above and below for finite weights due to their bounded activation functions, guaranteeing passive behavior.

The authors note that hybrid models might still require richer candidate‑function libraries—such as Gaussian functions—to achieve low prediction error, especially when noise is present.