  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.
