  We investigate the optimal transport problem between probability measures
when the underlying cost function is understood to satisfy a least action
principle, also known as a Lagrangian cost. These generalizations are useful
when connecting observations from a physical system where the transport
dynamics are influenced by the geometry of the system, such as obstacles (e.g.,
incorporating barrier functions in the Lagrangian), and allows practitioners to
incorporate a priori knowledge of the underlying system such as non-Euclidean
geometries (e.g., paths must be circular). Our contributions are of
computational interest, where we demonstrate the ability to efficiently compute
geodesics and amortize spline-based paths, which has not been done before, even
in low dimensional problems. Unlike prior work, we also output the resulting
Lagrangian optimal transport map without requiring an ODE solver. We
demonstrate the effectiveness of our formulation on low-dimensional examples
taken from prior work. The source code to reproduce our experiments is
available at https://github.com/facebookresearch/lagrangian-ot.
