Pseudo-Hamiltonian system identification

TL;DR

The paper tackles the challenge of identifying dynamical systems from data when external disturbances and damping obscure the internal dynamics. It introduces pseudo-Hamiltonian system identification (PHSI), learning the internal Hamiltonian $H$, damping $R$, and external forces $F$ (with a neural network for $F$ in a hybrid model) under a known structure matrix $S$ and a loss based on a fourth-order symmetric integrator: $\hat{g}_{\theta}(x,t) = (\hat{S}_{\theta}(x) - \hat{R}_{\theta}(x)) \nabla \hat{H}_{\theta}(x) + \hat{F}_{\theta}(x,t)$. Key contributions include pruning and $L_1$ regularization to promote sparsity and identifiability, robust performance on noisy data, and the ability to separate internal dynamics from external forces across separable, non-separable, and pseudo-Hamiltonian systems, with demonstrations on mass–spring, Hénon–Heiles, nonlinear Schrödinger, and tank-leak networks. The approach yields improved extrapolation and interpretability over baseline system identification methods and is supported by open-source code, enabling broader application to realistic dynamical problems. Overall, PHSI broadens the applicability of physics-informed structure learning to complex, real-world systems where disturbances cannot be neglected.

Abstract

Identifying the underlying dynamics of physical systems can be challenging when only provided with observational data. In this work, we consider systems that can be modelled as first-order ordinary differential equations. By assuming a certain pseudo-Hamiltonian formulation, we are able to learn the analytic terms of internal dynamics even if the model is trained on data where the system is affected by unknown damping and external disturbances. In cases where it is difficult to find analytic terms for the disturbances, a hybrid model that uses a neural network to learn these can still accurately identify the dynamics of the system as if under ideal conditions. This makes the models applicable in some situations where other system identification models fail. Furthermore, we propose to use a fourth-order symmetric integration scheme in the loss function and avoid actual integration in the training, and demonstrate on varied examples how this leads to increased performance on noisy data.

Figures (21)

• Figure 1: Comparison of simulated trajectories for the Hénon--Heiles system, with initial value $(q,p) = (0.1, -0.2, 0.4, 0.5)$, from $t=0$ to $t=10$.
• Figure 2: The average $L_2$-error of the trajectories obtained with 10 different random initial conditions, from the different models of the Hénon--Heiles system trained on noisy data with $\sigma = 0.02$, compared to trajectories simulated from the exact system,
• Figure 3: Phase portraits showing the trained models' trajectories next to the ground truth trajectory, for the nonlinear Schrödinger system. All models are trained on the two training sets, one clean data set and one noisy with $\sigma= 10^{-4}$. The initial values are $(q,p) = (-0.3, 0.5, -0.2, -0.4)$.
• Figure 4: The trained coefficients of the PHSI, BSI, and SINDy models of the nonlinear Schrödinger system, plotted against the respective true values. A perfectly trained model will only have points along the dotted line. The models are trained on the noisy data set. Note that PHSI learns the Hamiltonian function while SINDy learns the right-hand side $g$ of \ref{['eq:ode']} and hence they will not learn the same coefficients for corresponding terms.
• Figure 5: Comparison between the phase portraits obtained from integrating the exact forced and damped mass-spring system and the learned models from time $0$ to $10$. The initial value is $(q,p) = (-3.4, -1.9)$.