  In this paper, we present a neural network approach to address the dynamic
unbalanced optimal transport problem on surfaces with point cloud
representation. For surfaces with point cloud representation, traditional
method is difficult to apply due to the difficulty of mesh generating. Neural
network is easy to implement even for complicate geometry. Moreover, instead of
solving the original dynamic formulation, we consider the Hamiltonian flow
approach, i.e. Karush-Kuhn-Tucker system. Based on this approach, we can
exploit mathematical structure of the optimal transport to construct the neural
network and the loss function can be simplified. Extensive numerical
experiments are conducted for surfaces with different geometry. We also test
the method for point cloud with noise, which shows stability of this method.
This method is also easy to generalize to diverse range of problems.
