  We propose 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
is 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
coupling density. Furthermore, the proposed scheme can be used to describe the
behaviors of Hamilton-Jacobi PDEs beyond the singularity formations on the
support of coupling density.Several numerical examples with different
Hamiltonians are provided to support our findings.
