cs.LGMar 10, 2026

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

Authors: Luca Di Persio, Matthias Ehrhardt, Youness Outaleb

Organizations: Department of Computer Science - College of Mathematics University of Verona, Strada le Grazie 15 - 37134 Verona, Italy · University of Wuppertal, Chair of Applied and Computational Mathematics, Gaußstrasse 20, 42119 Wuppertal, Germany · Doctoral School in Mathematics, University of Trento Via Sommarive, 14 - 38123 Povo, Italy

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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 C2C^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 & tables

VariantObjective
SPH-NN-IBIncrement-based loss LIB\mathcal{L}_{\text{IB}}
SPH-NN-CEConditional-expectation loss LCE\mathcal{L}_{\text{CE}}
SPH-NN-NLLEuler-Maruyama neg. log-likelihood LNLL\mathcal{L}_{\text{NLL}}
Table 1: SPH-NN variants trained with three objectives.
SystemModelTrue-MSEMean ∣Δq∣|\Delta q|Mean ∣Δp∣|\Delta p|Mean ∣ΔH∣|\Delta H|
Mass-springBaseline3.63×10−33.63\times 10^{-3}0.2590.2690.557
Mass-springSPH-NN-IB2.68×10−32.68\times 10^{-3}0.0650.0490.065
Mass-springSPH-NN-CE1.14×10−31.14\times 10^{-3}0.0590.0570.011
Mass-springSPH-NN-NLL2.65×10−32.65\times 10^{-3}0.0650.0520.055
DuffingBaseline5.67×10−35.67\times 10^{-3}0.3310.3930.083
DuffingSPH-NN-IB2.60×10−32.60\times 10^{-3}0.0180.0170.005
Table 2: One-step drift error (True-MSE) and mean absolute rollout errors on the deterministic mean dynamics over t∈[0,20]t\in[0,20] from (q0,p0)=(1,0)(q_{0},p_{0})=(1,0) . The energy error uses the ground-truth Hamiltonian HH .
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.
Figure 6: Phase space - Duffing oscillator: the baseline spirals inward, while SPH methods stay close to the true closed trajectory.
Figure 7: Mean squared error of the rollout - Duffing oscillator.
Figure 8: Energy error - Duffing oscillator.
Figure 9: Energy evolution - Duffing oscillator.
Figure 10: Position and momentum evolution - Duffing oscillator.
Figure 11: Phase space - Van der Pol.
Figure 12: Mean squared error of the rollout - Van der Pol: the baseline error explodes with large spikes, while SPH methods keep the error low and stable across the rollout.
Figure 13: Energy error - Van der Pol oscillator.
Figure 14: Energy evolution - Van der Pol oscillator: SPH methods match the true energy oscillations closely, while the baseline slowly drifts and becomes misaligned.
Figure 15: Position and momentum evolution - Van der Pol oscillator.

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