  A method for learning Hamiltonian dynamics from a limited and noisy dataset
is proposed. The method learns a Hamiltonian vector field on a reproducing
kernel Hilbert space (RKHS) of inherently Hamiltonian vector fields, and in
particular, odd Hamiltonian vector fields. This is done with a symplectic
kernel, and it is shown how the kernel can be modified to an odd symplectic
kernel to impose the odd symmetry. A random feature approximation is developed
for the proposed kernel to reduce the problem size. This includes random
feature approximations for odd kernels. The performance of the method is
validated in simulations for three Hamiltonian systems. It is demonstrated that
the use of an odd symplectic kernel improves prediction accuracy, and that the
learned vector fields are Hamiltonian and exhibit the imposed odd symmetry
characteristics.
