Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees

TL;DR

This paper introduces stochastic port-Hamiltonian neural networks (SPH-NNs) that model open stochastic systems with energy-based structure, providing universal approximation and passivity guarantees, and demonstrating improved performance on oscillatory systems.

Contribution

The paper proposes SPH-NNs that incorporate physical structure and passivity into neural networks, with proven universal approximation and stability properties.

Findings

SPH-NNs approximate stochastic port-Hamiltonian systems with high accuracy.

SPH-NNs exhibit improved long-term stability in simulations.

Energy errors are reduced compared to baseline neural networks.

Abstract

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 It\^o 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.