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：我们提出了一种用于高维一阶哈密顿-雅可比偏微分方程的监督学习方案。该方案利用Wasserstein哈密顿流在密度耦合策略下的几何结构进行设计。通过Bregman散度，它被等价地转化为一个回归问题，该散度提供了学习过程中的损失函数，而数据则通过Wasserstein哈密顿流的粒子公式生成。我们基于耦合密度证明了所提方案的$L^1$残差的后验估计。此外，该方案可用于描述耦合密度支撑上奇点形成之外的哈密顿-雅可比偏微分方程行为。我们提供了多个不同哈密顿量的数值示例来支持我们的发现。