Learning momenta for Hamiltonian-based models

Ignacio Puiggros T.; A. Srikantha Phani

cs.LGOct 31, 2026

Learning momenta for Hamiltonian-based models

Authors: Ignacio Puiggros T., A. Srikantha Phani

Organizations: Department of Mechanical Engineering, The University of British Columbia, 2054-6250 Applied Science Lane, Vancouver, V6T 1Z4, British Columbia, Canada

Abstract

There has been increasing interest in methodologies that incorporate physics priors into neural network architectures to enhance their modeling capabilities. Among these methodologies, Hamiltonian neural networks (HNNs) and their variants have gained considerable attention. These architectures explicitly encode Hamiltonian mechanics both in their structure and loss function, resulting in desirable properties and additional modeled information. A crucial limitation of HNNs is their reliance on data on the target system's generalized momenta. This severely hampers their applicability when the analytical form of the generalized momenta is unknown, which frequently is the case in real-world applications. The goal of the present work is to develop a general extension of Hamiltonian-based architectures that addresses this issue by simultaneously learning the system's generalized momenta, Hamiltonian structure, and dynamics from data. The proposed architecture learns the generalized momenta when their analytical form is unknown, and only (potentially non-sequential) data on the state and its derivatives is available. A dissipative mass-spring system, a double pendulum, and a rolling disk under nonholonomic constraints are considered as canonical examples to assess the performance of the proposed architecture.

Figures & tables

Figure 1: Network architectures incorporating the proposed extension. Solid arrows represent explicit network inputs and outputs. a) Two-network architecture to learn the Hamiltonian and momenta of conservative, unconstrained systems. b) Hamiltonian NN architecture for modeling systems under nonholonomic constraints extended to learn generalized momenta from data. c) Architecture used to model the parameter-dependent dissipative mass-spring system. Dashed lines indicate quantities used to construct the Hamiltonian. $k_{c}$ and $d_{2}$ are parameters of the system.

Figure 2: a) Notations and coordinates of the chosen dissipative mass-spring system. b) Choice of coordinates for the double pendulum. c) Disk rolling without slipping. The coordinates $q_{1}$ and $q_{2}$ indicate the horizontal position of the center of the disk, $q_{4}$ is the angle of rotation with respect to the vertical axis, and $q_{5}$ represents the rolling angle of the disk about its main symmetry axis. $R$ is the radius of the disk. d) $q_{3}$ is the leaning angle of the disk from its fully vertical position.

Figure 3: Evolution of the training loss function for a) the mass spring system, b) the double pendulum, and c) the rolling disk.

$m_{1}$	$m_{2}$	$k_{1}$	$k_{2}$	$k_{c}$	$c_{1}$	$c_{2}$	$c_{c}$
$2$	$1$	$1$	$1$	$2.41421$	$0$	$2.61313$	$0$

Table 1: Parameters associated with the chosen non-Hermitian degeneracy of the mass-spring system.

Figure 4: Modeled and ground truth trajectories for the test systems. A family of test trajectories in coordinate space of a) the mass-spring system and b) the double pendulum. c) Trajectory in space of the center of the disk and d) orientation angles of the rolling disk.

Figure 5: Phase shift as a function of the circling radius in parameter space, for one and two cycles.

Figure 6: Comparison between learned and analytical generalized momenta for all systems. Different colors correspond to different components of the momenta, and every component, with the exception of the angular components of the rolling disk, was centered around 0 and scaled to unit mean absolute value for ease of visualization. First row shows momenta along the test trajectories for a) the mass-spring system, b) the double pendulum, c) spatial components of the rolling disk

[p_{1},p_{2}]

, and d) angular components of the rolling disk

[p_{3},p_{4},p_{5}]

. Angular components of the rolling disk are plotted unmodified when constant along the trajectory for ease of visualization. The second row compares the analytical momenta along the test trajectory with the learned momenta after being transformed as in Eq. 15 . Panel e) corresponds to the mass-spring system, f) to the double pendulum, and g) and h) to the spatial and angular components of the rolling disk momenta, respectively.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	$\mathbf{p}$	$\mathbf{K}$ and $\mathbf{D}$	$\mathbf{M}$
Hidden layers	2	2	N/A
Neurons per layer	30	10	N/A
Activation function	Tanh	Tanh	N/A
Learning rate	$1\times 10^{-5}$	$1\times 10^{-4}$	$1\times 10^{-4}$
Epochs	5e4
Training set	5e4 points

Appendix

Table 1: Training parameters used for the mass-spring experiment.

Parameter	Value
Hidden layers	2
Neurons per layer	30
Activation function	Tanh
Learning rate	5e-5
Epochs	3e6
Training set	4e3 points

Appendix

Table 2: Parameters used for the double pendulum experiment.

Parameter	$H$	$\bm{\lambda}$	A	p
Hidden layers	2	2	2	2
Neurons per layer	400	400	100	400
Activation function	Tanh	Tanh	Tanh	Tanh
Learning rate	1e-5	1e-5	1e-5	1e-5
Epochs	6e4
Training set	4.9e4 points

Appendix

Table 3: Parameters used for the rolling disk experiment.

Coordinate	Lower limit	Upper limit
$q_{1}$	$-2.3$	$2.3$
$q_{2}$	$-2.3$	$2.3$
$q_{3}$	$0.3$	$0.5$
$q_{4}$	$-0.3$	$7$
$q_{5}$	$-0.3$	$16$
$\dot{q}_{1}$	$-2.6$	$2.6$

Appendix

Table 4: Ranges of values for generated points for the rolling disk experiment.

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