  We are interested in learning generative models for complex geometries
described via manifolds, such as spheres, tori, and other implicit surfaces.
Current extensions of existing (Euclidean) generative models are restricted to
specific geometries and typically suffer from high computational costs. We
introduce Moser Flow (MF), a new class of generative models within the family
of continuous normalizing flows (CNF). MF also produces a CNF via a solution to
the change-of-variable formula, however differently from other CNF methods, its
model (learned) density is parameterized as the source (prior) density minus
the divergence of a neural network (NN). The divergence is a local, linear
differential operator, easy to approximate and calculate on manifolds.
Therefore, unlike other CNFs, MF does not require invoking or backpropagating
through an ODE solver during training. Furthermore, representing the model
density explicitly as the divergence of a NN rather than as a solution of an
ODE facilitates learning high fidelity densities. Theoretically, we prove that
MF constitutes a universal density approximator under suitable assumptions.
Empirically, we demonstrate for the first time the use of flow models for
sampling from general curved surfaces and achieve significant improvements in
density estimation, sample quality, and training complexity over existing CNFs
on challenging synthetic geometries and real-world benchmarks from the earth
and climate sciences.
