  Geodesics are essential in many geometry processing applications. However,
traditional algorithms for computing geodesic distances and paths on 3D mesh
models are often inefficient and slow. This makes them impractical for
scenarios that require extensive querying of arbitrary point-to-point
geodesics. Although neural implicit representations have emerged as a popular
way of representing 3D shape geometries, there is still no research on
representing geodesics with deep implicit functions. To bridge this gap, this
paper presents the first attempt to represent geodesics on 3D mesh models using
neural implicit functions. Specifically, we introduce neural geodesic fields
(NeuroGFs), which are learned to represent the all-pairs geodesics of a given
mesh. By using NeuroGFs, we can efficiently and accurately answer queries of
arbitrary point-to-point geodesic distances and paths, overcoming the
limitations of traditional algorithms. Evaluations on common 3D models show
that NeuroGFs exhibit exceptional performance in solving the single-source
all-destination (SSAD) and point-to-point geodesics, and achieve high accuracy
consistently. Besides, NeuroGFs also offer the unique advantage of encoding
both 3D geometry and geodesics in a unified representation. Moreover, we
further extend generalizable learning frameworks of NeuroGFs by adding shape
feature encoders, which also show satisfactory performances for unseen shapes
and categories. Code is made available at
https://github.com/keeganhk/NeuroGF/tree/master.
