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.

翻译：本文提出了一种针对高维一阶Hamilton-Jacobi偏微分方程的监督学习方案。该方案通过密度耦合策略，利用Wasserstein哈密顿流的几何结构进行设计。该问题可等价地表述为使用Bregman散度的回归问题，从而为学习过程提供损失函数，而数据则通过Wasserstein哈密顿流的粒子化表述生成。我们基于耦合密度证明了所提方案在$L^1$残差上的后验估计。此外，所提方案可用于描述Hamilton-Jacobi偏微分方程在耦合密度支撑集上超越奇点形成的行为。文中提供了多个具有不同哈密顿量的数值算例以支持我们的发现。