  Concise, accurate descriptions of physical systems through their conserved
quantities abound in the natural sciences. In data science, however, current
research often focuses on regression problems, without routinely incorporating
additional assumptions about the system that generated the data. Here, we
propose to explore a particular type of underlying structure in the data:
Hamiltonian systems, where an "energy" is conserved. Given a collection of
observations of such a Hamiltonian system over time, we extract phase space
coordinates and a Hamiltonian function of them that acts as the generator of
the system dynamics. The approach employs an autoencoder neural network
component to estimate the transformation from observations to the phase space
of a Hamiltonian system. An additional neural network component is used to
approximate the Hamiltonian function on this constructed space, and the two
components are trained jointly. As an alternative approach, we also demonstrate
the use of Gaussian processes for the estimation of such a Hamiltonian. After
two illustrative examples, we extract an underlying phase space as well as the
generating Hamiltonian from a collection of movies of a pendulum. The approach
is fully data-driven, and does not assume a particular form of the Hamiltonian
function.
