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

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 $\mathbf{J}$ corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix $\mathbf{R}$ 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.

In this work, we extend the Equilibrium Propagation framework to skew-gradient systems and show an equivalence between deep Energy-Based Models and Hamiltonian neural networks. We focus on networks of diffusively coupled Fitzhugh-Nagumo neurons as a prototypical example. We show that since stationary solutions of the Fitzhugh-Nagumo model are described by self-adjoint operators, the methods of equilibrium propagation for performing credit assignment can be applied. Furthermore, for Fitzhugh-Nagumo networks with the topology of a deep residual network, we show that the steady state solutions admit a (spatial) Hamiltonian, and thus the methods of Hamiltonian Echo Backpropagation can be applied. We end by deriving an explicit layer-wise Hamiltonian recurrence relation governing inference for stationary solutions of both deep Fitzhugh-Nagumo networks and deep Energy-Based Models.

Learning neural-network models of dynamical systems with safety guarantees is a fundamental requirement for their deployment in safety-critical settings. Safety is commonly established by proving the invariance of a desired subset in state-space, ensuring that every trajectory initialized in this subset remains confined to it for all time under admissible inputs. Existing frameworks, however, either rely on computationally expensive post-hoc verification or employ safety-enforcing mechanisms without formal correctness guarantees. In this paper, we introduce a novel neural architecture grounded in energy-based modern Hopfield networks to guarantee safety-by-design while retaining sufficient expressiveness to model complex nonlinear dynamics. Specifically, we integrate modern Hopfield networks with a port-Hamiltonian neural ODE, enabling by design the construction of barrier functions yielding explicit admissible-input sets and quantitative robustness radii. Across several benchmarks, including an 12-dimensional nanodrone model, our framework achieves state-of-the-art performance while producing certified invariant sets that are more robust to external solicitations than comparable existing approaches.