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

TL;DR

Stochastic port-Hamiltonian neural networks, SPH-NNs, are introduced, which parameterize the Hamiltonian with a feedforward network and enforce skew symmetry of the interconnection matrix and positive semidefiniteness of the dissipation matrix.

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 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.

Figures (15)

• Figure 1: Phase space - Mass spring oscillator.
• Figure 2: Mean squared error of the rollout - Mass-spring oscillator: the baseline error grows quickly, while the SPH methods keep the error low and stable.
• Figure 3: Energy error - Mass-spring oscillator: SPH-NN models significantly reduce energy error compared to the baseline, with the CE loss achieving the most accurate and stable energy behavior over time.
• Figure 4: Energy evolution - Mass-spring oscillator: the baseline loses energy steadily, while SPH models keep energy close to the true constant value.
• Figure 5: Position and momentum time evolution - Mass-spring oscillator: the baseline drifts and loses amplitude, while SPH methods track the true oscillation and keep errors much smaller.