A scalable deep learning approach for solving high-dimensional dynamic optimal transport

Wei Wan School of mathematics and physics, North China Electric Power University Yuejin Zhang Department of Mathematical Science, Tsinghua University Chenglong Bao Yau Mathematical Sciences Center, Tsinghua University, and Yaqi Lake Beijing Institute of Mathematical Sciences and Applications Bin Dong Beijing International Center for Mathematical Research, Peking University, and Center for Ma- chine Learning Research, Peking University Zuoqiang Shi Yau Mathematical Sciences Center, Tsinghua University, and Yaqi Lake Beijing Institute of Mathematical Sciences and Applications

Machine Learning mathscidoc:2206.41001

2022.5
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.
No keywords uploaded!
[ Download ] [ 2022-06-13 16:46:11 uploaded by Baocl ] [ 47 downloads ] [ 0 comments ]
@inproceedings{wei2022a,
  title={A scalable deep learning approach for solving high-dimensional dynamic optimal transport},
  author={Wei Wan, Yuejin Zhang, Chenglong Bao, Bin Dong, and Zuoqiang Shi},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613164611086929346},
  year={2022},
}
Wei Wan, Yuejin Zhang, Chenglong Bao, Bin Dong, and Zuoqiang Shi. A scalable deep learning approach for solving high-dimensional dynamic optimal transport. 2022. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220613164611086929346.
Please log in for comment!
 
 
Contact us: office-iccm@tsinghua.edu.cn | Copyright Reserved