  A fundamental result of thermodynamic geometry is that the optimal,
minimal-work protocol that drives a nonequilibrium system between two
thermodynamic states in the slow-driving limit is given by a geodesic of the
friction tensor, a Riemannian metric defined on control space. For overdamped
dynamics in arbitrary dimensions, we demonstrate that thermodynamic geometry is
equivalent to $L^2$ optimal transport geometry defined on the space of
equilibrium distributions corresponding to the control parameters. We show that
obtaining optimal protocols past the slow-driving or linear response regime is
computationally tractable as the sum of a friction tensor geodesic and a
counterdiabatic term related to the Fisher information metric. These
geodesic-counterdiabatic optimal protocols are exact for parameteric harmonic
potentials, reproduce the surprising non-monotonic behavior recently discovered
in linearly-biased double well optimal protocols, and explain the ubiquitous
discontinuous jumps observed at the beginning and end times.
