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来计算Wasserstein哈密顿流（WHF），其定义在概率密度流形上的哈密顿系统。许多著名的偏微分方程系统均可重新表述为WHF。我们采用参数化函数作为推进映射来刻画WHF的解，并将偏微分方程转化为有限维常微分方程系统——这一系统在参数流形的相空间中构成哈密顿系统。我们建立了连续时间逼近方案在Wasserstein度量下的误差分析结果。数值实现中，我们使用神经网络作为推进映射，并采用高效的辛格式求解导出的哈密顿常微分方程系统，使方法能够保持总能量等重要物理量。整个计算过程通过完全确定性的辛积分器完成，无需任何神经网络训练。因此，本方法不涉及网络参数的直接优化，从而避免了通常难以量化评估的随机梯度下降（SGD）方法引入的误差。该算法是一种基于采样的方法，能有效扩展至高维问题。此外，本方法还通过参数化常微分方程动力学，为原始WHF的拉格朗日视角与欧拉视角之间的关联提供了新的途径。