  Maps are arguably one of the most fundamental concepts used to define and
operate on manifold surfaces in differentiable geometry. Accordingly, in
geometry processing, maps are ubiquitous and are used in many core
applications, such as paramterization, shape analysis, remeshing, and
deformation. Unfortunately, most computational representations of surface maps
do not lend themselves to manipulation and optimization, usually entailing
hard, discrete problems. While algorithms exist to solve these problems, they
are problem-specific, and a general framework for surface maps is still in
need. In this paper, we advocate considering neural networks as encoding
surface maps. Since neural networks can be composed on one another and are
differentiable, we show it is easy to use them to define surfaces via atlases,
compose them for surface-to-surface mappings, and optimize differentiable
objectives relating to them, such as any notion of distortion, in a trivial
manner. In our experiments, we represent surfaces by generating a neural map
that approximates a UV parameterization of a 3D model. Then, we compose this
map with other neural maps which we optimize with respect to distortion
measures. We show that our formulation enables trivial optimization of rather
elusive mapping tasks, such as maps between a collection of surfaces.
