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The dynamic formulation of optimal transport has attracted growing interest in scientific computing and machine learning, and its computation requires solving a PDE-constrained optimization problem. The classical Eulerian discretization based approaches suffer from the curse of dimensionality, which arises from the approximation of a high-dimensional velocity field. In this work, we propose a deep learning based method for solving the dynamic optimal transport in highdimensional space. Our method contains three main ingredients: a carefully designed representation of the velocity field, the discretization of the PDE constraint along the characteristics, and the computation of a high-dimensional integral by the Monte Carlo method in each time step. Specifically, in the representation of the velocity field, we apply the classical nodal basis function in time and the deep neural networks in the space domain with the H1-norm regularization. This technique promotes the regularity of the velocity field in both time and space such that the discretization along the characteristic remains stable during the training process. Extensive numerical examples have been conducted to test the proposed method. Compared to other solvers of optimal transport, our method could give more accurate results in high-dimensional cases and has very good scalability with respect to dimension. Finally, we extend our method to more complicated cases such as the crowd motion problem. [ABSTRACT FROM AUTHOR]