Stochastic Port-Hamiltonian Neural Networks: Universal Approximation with Passivity Guarantees
arXiv:2603.10078v1 Announce Type: new 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 ro
arXiv:2603.10078v1 Announce Type: new 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.
Executive Summary
This article introduces stochastic port-Hamiltonian neural networks (SPH-NNs) as a novel approach to modeling open dynamical systems with dissipation, inputs, and stochastic forcing. By parameterizing the Hamiltonian with a feedforward network and enforcing key matrix properties, SPH-NNs establish a weak passivity inequality and demonstrate universal approximation capabilities. Experimental results on noisy oscillators showcase improved performance compared to a multilayer perceptron baseline. This work offers a promising framework for modeling and analysis of complex systems, particularly in the context of control and energy-based systems. However, further investigation into the scalability and interpretability of SPH-NNs is warranted.
Key Points
- ▸ SPH-NNs parameterize the Hamiltonian with a feedforward network to model open dynamical systems
- ▸ Enforce skew symmetry and positive semidefiniteness of key matrices for passivity guarantees
- ▸ Establish weak passivity inequality and universal approximation capabilities for stochastic port-Hamiltonian systems
Merits
Strength in Mathematical Rigor
The article demonstrates a high level of mathematical rigor, particularly in establishing the passivity inequality and universal approximation results. The use of Ito dynamics and explicit generator conditions adds depth to the analysis.
Practical Application
Experimental results on noisy oscillators showcase the potential of SPH-NNs in modeling real-world systems, offering a promising framework for control and energy-based systems.
Demerits
Scalability Limitation
The article does not address scalability concerns, which may limit the practical application of SPH-NNs in more complex systems.
Interpretability Concerns
The use of feedforward networks to parameterize the Hamiltonian may raise interpretability concerns, making it challenging to understand the underlying dynamics of the system.
Expert Commentary
The introduction of stochastic port-Hamiltonian neural networks represents a significant advancement in the field of control theory and energy-based systems. By leveraging the strengths of both port-Hamiltonian systems and neural networks, SPH-NNs offer a novel approach to modeling complex systems. However, further research is needed to address scalability and interpretability concerns. The experimental results presented in the article demonstrate the potential of SPH-NNs in real-world applications, making this a promising area of research for control engineers, mathematicians, and policymakers.
Recommendations
- ✓ Investigate the scalability of SPH-NNs in more complex systems to further establish their practical application.
- ✓ Explore the use of SPH-NNs in conjunction with other control strategies to develop more advanced control approaches.
