The modelling of dynamical systems from discrete observations is a challenge
faced by modern scientific and engineering data systems. Hamiltonian systems
are one such fundamental and ubiquitous class of dynamical systems. Hamiltonian
neural networks are state-of-the-art models that unsupervised-ly regress the
Hamiltonian of a dynamical system from discrete observations of its vector
field under the learning bias of Hamilton's equations. Yet Hamiltonian dynamics
are often complicated, especially in higher dimensions where the state space of
the Hamiltonian system is large relative to the number of samples. A recently
discovered remedy to alleviate the complexity between state variables in the
state space is to leverage the additive separability of the Hamiltonian system
and embed that additive separability into the Hamiltonian neural network.
Following the nomenclature of physics-informed machine learning, we propose
three separable Hamiltonian neural networks. These models embed additive
separability within Hamiltonian neural networks. The first model uses additive
separability to quadratically scale the amount of data for training Hamiltonian
neural networks. The second model embeds additive separability within the loss
function of the Hamiltonian neural network. The third model embeds additive
separability through the architecture of the Hamiltonian neural network using
conjoined multilayer perceptions. We empirically compare the three models
against state-of-the-art Hamiltonian neural networks, and demonstrate that the
separable Hamiltonian neural networks, which alleviate complexity between the
state variables, are more effective at regressing the Hamiltonian and its
vector field.Comment: 11 page