Port-Hamiltonian Neural Networks: From Theory to Simulation of Interconnected Stochastic Systems

This work introduces a new framework integrating port-Hamiltonian systems (PHS) and neural network architectures. This framework bridges the gap between deterministic and stochastic modeling of complex dynamical systems. We introduce new mathematical formulations and computational methods that expand the geometric structure of PHS to account for uncertainty, environmental noise, and random perturbations. Building on these advances, we introduce stochastic port-Hamiltonian neural networks (pHNNs), which facilitate the accurate learning and prediction of non-autonomous and interconnected stochastic systems. Our proposed framework generalizes passivity concepts to the stochastic regime, ensuring stability while maintaining the system's energy-consistent structure. Extensive simulations, including those involving damped mass-spring systems, Duffing oscillators, and robotic control tasks, demonstrate the capability of pHNNs to capture complex dynamics with high fidelity, even under noise and uncertainty. This unified approach establishes a foundation for the robust, data-driven modeling and control of nonlinear stochastic systems.