  Adaptive control is a critical component of reliable robot autonomy in
rapidly changing operational conditions. Adaptive control designs benefit from
a disturbance model, which is often unavailable in practice. This motivates the
use of machine learning techniques to learn disturbance features from training
data offline, which can subsequently be employed to compensate the disturbances
online. This paper develops geometric adaptive control with a learned
disturbance model for rigid-body systems, such as ground, aerial, and
underwater vehicles, that satisfy Hamilton's equations of motion over the
$SE(3)$ manifold. Our design consists of an \emph{offline disturbance model
identification stage}, using a Hamiltonian-based neural ordinary differential
equation (ODE) network trained from state-control trajectory data, and an
\emph{online adaptive control stage}, estimating and compensating the
disturbances based on geometric tracking errors. We demonstrate our adaptive
geometric controller in trajectory tracking simulations of fully-actuated
pendulum and under-actuated quadrotor systems.
