We propose two graph neural network layers for graphs with features in a
Riemannian manifold. First, based on a manifold-valued graph diffusion
equation, we construct a diffusion layer that can be applied to an arbitrary
number of nodes and graph connectivity patterns. Second, we model a tangent
multilayer perceptron by transferring ideas from the vector neuron framework to
our general setting. Both layers are equivariant with respect to node
permutations and isometries of the feature manifold. These properties have been
shown to lead to a beneficial inductive bias in many deep learning tasks.
Numerical examples on synthetic data as well as on triangle meshes of the right
hippocampus to classify Alzheimer's disease demonstrate the very good
performance of our layers.

基于黎曼流形的图神经网络模型中，我们提出了两个关键的图神经网络层。第一个是扩散层，其基于流形值图扩散方程，适用于任意数量节点和图的连通模式。第二个是切线多层感知机层，借鉴了向量神经元框架的思想，并在一般的情境中应用。这两个层在节点排列和特征流形的等变性方面表现出非常好的性能。在合成数据和海马右侧三角网格对阿尔茨海默病分类的数值实例中，我们的模型均取得了非常好的性能。