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.