  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.
