A Supervised Learning Scheme for Computing Hamilton–Jacobi Equation via Density Coupling

Abstract. 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 [Formula: see text] 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.