We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space embeddings of probability measures. We prove that the transport maps given by the proposed methods converge to optimal transport maps in the problem with $L^2$ cost. Several numerical experiments validate our methods. In particular, we show that our methods are applicable to large-scale Monge problems.

翻译：针对经典蒙日最优质量传输问题，我们提出了一种深度学习方法，其中分布约束通过希尔伯特空间概率测度嵌入理论中的最大均值差异作为惩罚项进行处理。我们证明了所提方法给出的传输映射在具有$L^2$代价的问题中收敛于最优传输映射。多项数值实验验证了该方法的有效性，特别展示了该方法适用于大规模蒙日问题的求解。