  We present GeGnn, a learning-based method for computing the approximate
geodesic distance between two arbitrary points on discrete polyhedra surfaces
with constant time complexity after fast precomputation. Previous relevant
methods either focus on computing the geodesic distance between a single source
and all destinations, which has linear complexity at least or require a long
precomputation time. Our key idea is to train a graph neural network to embed
an input mesh into a high-dimensional embedding space and compute the geodesic
distance between a pair of points using the corresponding embedding vectors and
a lightweight decoding function. To facilitate the learning of the embedding,
we propose novel graph convolution and graph pooling modules that incorporate
local geodesic information and are verified to be much more effective than
previous designs. After training, our method requires only one forward pass of
the network per mesh as precomputation. Then, we can compute the geodesic
distance between a pair of points using our decoding function, which requires
only several matrix multiplications and can be massively parallelized on GPUs.
We verify the efficiency and effectiveness of our method on ShapeNet and
demonstrate that our method is faster than existing methods by orders of
magnitude while achieving comparable or better accuracy. Additionally, our
method exhibits robustness on noisy and incomplete meshes and strong
generalization ability on out-of-distribution meshes. The code and pretrained
model can be found on https://github.com/IntelligentGeometry/GeGnn.
