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.

利用一类神经背景成本函数的蒙热映射已知形式，我们构建了一个可微分的蒙热映射估计器，其可以根据已知信息优化，从而适应于特定问题的最优输运映射和相应背景成本函数的学习。该方法通过合适的损失函数，提供了一种利用关于蒙热映射本身的先前信息来学习适应于最优输运映射和背景成本函数的通用方法。

可微分的成本参数化蒙特映射估计