  Within the field of optimal transport (OT), the choice of ground cost is
crucial to ensuring that the optimality of a transport map corresponds to
usefulness in real-world applications. It is therefore desirable to use known
information to tailor cost functions and hence learn OT maps which are adapted
to the problem at hand. By considering a class of neural ground costs whose
Monge maps have a known form, we construct a differentiable Monge map estimator
which can be optimized to be consistent with known information about an OT map.
In doing so, we simultaneously learn both an OT map estimator and a
corresponding adapted cost function. Through suitable choices of loss function,
our method provides a general approach for incorporating prior information
about the Monge map itself when learning adapted OT maps and cost functions.
