  We present a novel approximate inference method for diffusion processes,
based on the Wasserstein gradient flow formulation of the diffusion. In this
formulation, the time-dependent density of the diffusion is derived as the
limit of implicit Euler steps that follow the gradients of a particular free
energy functional. Existing methods for computing Wasserstein gradient flows
rely on discretization of the domain of the diffusion, prohibiting their
application to domains in more than several dimensions. We propose instead a
discretization-free inference method that computes the Wasserstein gradient
flow directly in a space of continuous functions. We characterize approximation
properties of the proposed method and evaluate it on a nonlinear filtering
task, finding performance comparable to the state-of-the-art for filtering
diffusions.
