Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees
This work addresses the challenge of accurately simulating stochastic physical systems with guarantees for researchers in computational physics and machine learning, representing an incremental advancement by combining existing neural network and port-Hamiltonian frameworks.
The authors tackled the problem of modeling stochastic open dynamical systems by introducing stochastic port-Hamiltonian neural networks (SPH-NNs), which enforce energy-based properties like passivity and achieve universal approximation with C^2 accuracy, resulting in improved long horizon rollouts and reduced energy error compared to baseline methods in experiments on noisy oscillators.
Stochastic port-Hamiltonian systems represent open dynamical systems with dissipation, inputs, and stochastic forcing in an energy based form. We introduce stochastic port-Hamiltonian neural networks, SPH-NNs, which parameterize the Hamiltonian with a feedforward network and enforce skew symmetry of the interconnection matrix and positive semidefiniteness of the dissipation matrix. For Itô dynamics we establish a weak passivity inequality in expectation under an explicit generator condition, stated for a stopped process on a compact set. We also prove a universal approximation result showing that, on any compact set and finite horizon, SPH-NNs approximate the coefficients of a target stochastic port-Hamiltonian system with $C^2$ accuracy of the Hamiltonian and yield coupled solutions that remain close in mean square up to the exit time. Experiments on noisy mass spring, Duffing, and Van der Pol oscillators show improved long horizon rollouts and reduced energy error relative to a multilayer perceptron baseline.