In this talk, I will present a supervised learning scheme for the first order Hamilton--Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamiltonian flows via a density coupling strategy. It can be equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the particle formulation of Wasserstein Hamiltonian flow. We prove a posterior estimate on $L^1$ residual of the proposed scheme based on the support of
coupling density. Several numerical examples with different Hamiltonians are provided to support our findings. This presentation is based on a joint work with Jianbo Cui (HK PolyU) and Shu Liu (UCLA).
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