Structure- and Stability-Preserving Learning of Port-Hamiltonian Systems
Authors: Binh Nguyen, Nam T. Nguyen, Truong X. Nghiem
Organizations: Department of Electrical and Computer Engineering, College of Engineering and Computer Science, University of Central Florida, Orlando, FL 32816, USA
This paper investigates the problem of data-driven modeling of port-Hamiltonian systems while preserving their intrinsic Hamiltonian structure and stability properties. We propose a novel neural-network-based port-Hamiltonian modeling technique that relaxes the convexity constraint commonly imposed by neural network-based Hamiltonian approximations, thereby improving the expressiveness and generalization capability of the model. By removing this restriction, the proposed approach enables the use of more general non-convex Hamiltonian representations to enhance modeling flexibility and accuracy. Furthermore, the proposed method incorporates information about stable equilibria into the learning process, allowing the learned model to preserve the stability of multiple isolated equilibria rather than being restricted to a single equilibrium as in conventional methods. Two numerical experiments are conducted to validate the effectiveness of the proposed approach and demonstrate its ability to achieve more accurate structure- and stability-preserving learning of port-Hamiltonian systems compared with a baseline method.
Figures & tables
Fig. 1 : Illustration of h(σ(∥x∥2)) at xeq=0 .
Fig. 2 : Responses of the actual PHS, PH-ICNN, and proposed models to the sinusoid test signal for the Toda lattice system.
Fig. 3 : Responses of the actual PHS, PH-ICNN, and proposed models to the pulse test signal for the Toda lattice system.
Fig. 4 : A double pendulum system.
Fig. 5 : State trajectories of the double pendulum system from three different initial states x0(1)=[0,0,10,−2]⊤ , x0(2)=[0,0,1,−1]⊤ , and x0(3)=[0,0,−8,1]⊤ . These cases illustrate different stable equilibria xeq(1)=[2π,2π,0,0]⊤ , xeq(2)=[0,0,0,0]⊤ , and xeq(3)=[−2π,0,0,0]⊤ .
Stable port-Hamiltonian neural networks certify asymptotic stability by construction. Yet, their Hamiltonian is a global Lyapunov function with a single global minimum, so they can represent only dynamic systems with one attractor. We demonstrate that this excludes even simple systems with energy landscapes forming a double well, and we overcome the restriction by parametrising the Hamiltonian as a product of Bregman divergences generated by one input-convex network. We prove that the resulting model is locally Lyapunov stable, that the coexistence of stable equilibria forces additional non-asymptotically-stable equilibria to exist, that all equilibria lie in a bounded region, and under a hyperbolicity assumption that almost-everywhere stability holds. On three systems our approach is able to recover the energy surface characteristics and improve the convergence speed by 1.8×-8.5×.
Simon Heilig, Jens Püttschneider, Mohammad Itani +2
Faculty of Computer Science, Ruhr University Bochum, Germany · Institute of Control Systems, Hamburg University of Technology, Germany
Learning dynamical systems through purely data-driven methods is challenging as they do not learn the underlying conservation laws that enable them to correctly generalize. Existing port-Hamiltonian neural network methods have recently been successfully applied for modeling mechanical systems. However, even though these methods are designed on power-balance principles, they usually do not consider power-preserving discretizations and often rely on Runge-Kutta numerical methods. In this work, we propose to use a second-order discrete gradient method embedded in the learning of dynamical systems with port-Hamiltonian neural networks. Numerical results are provided for three systems deliberately selected to span different ranges of dynamical behavior under control: a baseline harmonic oscillator with quadratic energy storage; a Duffing oscillator, with a non-quadratic Hamiltonian offering amplitude-dependent effects; and a self-sustained oscillator, which can stabilize in a controlled limit cycle through the incorporation of a nonlinear dissipation. We show how the use of this discrete gradient method outperforms the performance of a Runge-Kutta method of the same order. Experiments are also carried out to compare two theoretically equivalent port-Hamiltonian systems formulations and to analyze the impact of regularizing the Jacobian of port-Hamiltonian neural networks during training.
Maximino Linares, Guillaume Doras, Thomas Hélie
IRCAM, 1, place Igor-Stravinsky, 75004, Paris, France
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 C2 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.
Luca Di Persio, Matthias Ehrhardt, Youness Outaleb
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