  The acceleration of gradient-based optimization methods is a subject of
significant practical and theoretical importance, particularly within machine
learning applications. While much attention has been directed towards
optimizing within Euclidean space, the need to optimize over spaces of
probability measures in machine learning motivates exploration of accelerated
gradient methods in this context too. To this end, we introduce a
Hamiltonian-flow approach analogous to momentum-based approaches in Euclidean
space. We demonstrate that, in the continuous-time setting, algorithms based on
this approach can achieve convergence rates of arbitrarily high order. We
complement our findings with numerical examples.
