We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture: namely, permutation equivariance, orientation equivariance, and simplicial awareness. The first two properties respectively account for the fact that the node indexing and the simplex orientations in a simplicial complex are arbitrary. The last property encodes the desirable feature that the output of the neural network depends on the entire simplicial complex and not on a subset of its dimensions. Based on these properties, we propose a simple convolutional architecture, rooted in tools from algebraic topology, for the problem of trajectory prediction, and show that it obeys all three of these properties when an odd, nonlinear activation function is used. We then demonstrate the effectiveness of this architecture in extrapolating trajectories on synthetic and real datasets, with particular emphasis on the gains in generalizability to unseen trajectories.
Hypergraph Neural Sheaf Diffusion: A Symmetric Simplicial Set Framework for Higher-Order Learning
POSTECH；Center for Geometry and Physics, Institute for Basic Science (IBS)（几何与物理中心，基础科学研究所）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling
Department of Computer Science and Engineering, Shanghai Jiao Tong University（上海交通大学计算机科学与工程系）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；School of Artificial Intelligence, Shanghai Jiao Tong University（上海交通大学人工智能学院）；SJTU Paris Elite Institute of Technology, Shanghai Jiao Tong University（上海交通大学巴黎精英理工学院）；Department of Physics, Harvard University（哈佛大学物理系）；Harvard-MIT Center for Ultracold Atoms（哈佛-麻省理工冷原子中心）；Eric and Wendy Schmidt Center, Broad Institute of MIT and Harvard（MIT和哈佛大学Broad研究所埃里克和wendy Schmidt中心）
