  Wasserstein gradient flows provide a powerful means of understanding and
solving many diffusion equations. Specifically, Fokker-Planck equations, which
model the diffusion of probability measures, can be understood as gradient
descent over entropy functionals in Wasserstein space. This equivalence,
introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme
to approximate these diffusion processes via an implicit discretization of the
gradient flow in Wasserstein space. Solving the optimization problem associated
to each JKO step, however, presents serious computational challenges. We
introduce a scalable method to approximate Wasserstein gradient flows, targeted
to machine learning applications. Our approach relies on input-convex neural
networks (ICNNs) to discretize the JKO steps, which can be optimized by
stochastic gradient descent. Unlike previous work, our method does not require
domain discretization or particle simulation. As a result, we can sample from
the measure at each time step of the diffusion and compute its probability
density. We demonstrate our algorithm's performance by computing diffusions
following the Fokker-Planck equation and apply it to unnormalized density
sampling as well as nonlinear filtering.
