In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use parameterized function as push-forward map to characterize the solution of WHF, and convert the PDE to a finite-dimensional ODE system, which is a Hamiltonian system in the phase space of the parameter manifold. We establish error analysis results for the continuous time approximation scheme in Wasserstein metric. For the numerical implementation, we use neural networks as push-forward maps. We apply an effective symplectic scheme to solve the derived Hamiltonian ODE system so that the method preserves some important quantities such as total energy. The computation is done by fully deterministic symplectic integrator without any neural network training. Thus, our method does not involve direct optimization over network parameters and hence can avoid the error introduced by stochastic gradient descent (SGD) methods, which is usually hard to quantify and measure. The proposed algorithm is a sampling-based approach that scales well to higher dimensional problems. In addition, the method also provides an alternative connection between the Lagrangian and Eulerian perspectives of the original WHF through the parameterized ODE dynamics.

翻译：本文提出了一种数值方法用于计算Wasserstein哈密顿流（Wasserstein Hamiltonian flow, WHF），该流是概率密度流形上的哈密顿系统。许多著名的偏微分方程（PDE）系统均可重构为WHF形式。我们采用参数化函数作为推前映射来刻画WHF的解，并将该偏微分方程转化为有限维常微分方程（ODE）系统，该系统是参数流形相空间中的哈密顿系统。我们建立了连续时间近似方案在Wasserstein度量下的误差分析结果。在数值实现中，我们使用神经网络作为推前映射，并采用有效的辛格式求解导出的哈密顿ODE系统，从而使该方法能够保持总能量等重要物理量。计算过程完全由确定性辛积分器完成，无需任何神经网络训练。因此，我们的方法不涉及对网络参数的直接优化，从而可避免随机梯度下降（SGD）方法引入的误差，此类误差通常难以量化和测量。所提算法是一种基于采样的方法，能较好地扩展至高维问题。此外，该方法还通过参数化ODE动力学，为原始WHF的拉格朗日视角与欧拉视角之间提供了另一种联系途径。