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.

本文探讨了建立基于单纯复形数据的神经网络体系结构。通过定义三种理想属性，即排列等变性，定向等变性和单纯感知性，我们提出了一种简单的卷积架构，用于轨迹预测问题，并展示了它在合成和实际数据集上推广性的提高。

基于原则的单纯神经网络用于轨迹预测