A scalable deep learning approach for solving high-dimensional dynamic
optimal transport
- URL: http://arxiv.org/abs/2205.07521v1
- Date: Mon, 16 May 2022 08:56:05 GMT
- Title: A scalable deep learning approach for solving high-dimensional dynamic
optimal transport
- Authors: Wei Wan, Yuejin Zhang, Chenglong Bao, Bin Dong, Zuoqiang Shi
- Abstract summary: We propose a deep learning based method to solve the dynamic optimal transport in high dimensional 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 high dimensional integral by Monte Carlo method in each time step.
- Score: 18.67654056717166
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: The dynamic formulation of optimal transport has attracted growing interests
in scientific computing and machine learning, and its computation requires to
solve a PDE-constrained optimization problem. The classical Eulerian
discretization based approaches suffer from the curse of dimensionality, which
arises from the approximation of high-dimensional velocity field. In this work,
we propose a deep learning based method to solve the dynamic optimal transport
in high dimensional 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 high
dimensional integral by 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 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 to be 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 crowd motion
problem.
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