Incorporating Euclidean symmetries (e.g. rotation equivariance) as inductive
biases into graph neural networks has improved their generalization ability and
data efficiency in unbounded physical dynamics modeling. However, in various
scientific and engineering applications, the symmetries of dynamics are
frequently discrete due to the boundary conditions. Thus, existing GNNs either
overlook necessary symmetry, resulting in suboptimal representation ability, or
impose excessive equivariance, which fails to generalize to unobserved
symmetric dynamics. In this work, we propose a general Discrete Equivariant
Graph Neural Network (DEGNN) that guarantees equivariance to a given discrete
point group. Specifically, we show that such discrete equivariant message
passing could be constructed by transforming geometric features into
permutation-invariant embeddings. Through relaxing continuous equivariant
constraints, DEGNN can employ more geometric feature combinations to
approximate unobserved physical object interaction functions. Two
implementation approaches of DEGNN are proposed based on ranking or pooling
permutation-invariant functions. We apply DEGNN to various physical dynamics,
ranging from particle, molecular, crowd to vehicle dynamics. In twenty
scenarios, DEGNN significantly outperforms existing state-of-the-art
approaches. Moreover, we show that DEGNN is data efficient, learning with less
data, and can generalize across scenarios such as unobserved orientation.

本文提出了一种基于离散等变的图神经网络（DEGNN），通过将几何特征转化为排列不变嵌入，实现相应离散点群的等变消息传递，从而提高对未观测到的对称动力学的表示能力和泛化性能。在各种物理动力学中，DEGNN 相较于现有的方法具有显著的优越性，并可以以较少的数据进行学习，并且可以跨领域进行泛化，如未观测到的方向。