  We propose two deep neural network-based methods for solving semi-martingale
optimal transport problems. The first method is based on a
relaxation/penalization of the terminal constraint, and is solved using deep
neural networks. The second method is based on the dual formulation of the
problem, which we express as a saddle point problem, and is solved using
adversarial networks. Both methods are mesh-free and therefore mitigate the
curse of dimensionality. We test the performance and accuracy of our methods on
several examples up to dimension 10. We also apply the first algorithm to a
portfolio optimization problem where the goal is, given an initial wealth
distribution, to find an investment strategy leading to a prescribed terminal
wealth distribution.
