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.

Graph Neural Networks (GNNs) have achieved notable success in various applications over graph data. However, recent research has revealed that real-world graphs often contain noise, and GNNs are susceptible to noise in the graph. To address this issue, several Graph Structure Learning (GSL) models have been introduced. While GSL models are tailored to enhance robustness against edge noise through edge reconstruction, a significant limitation surfaces: their high reliance on node features. This inherent dependence amplifies their susceptibility to noise within node features. Recognizing this vulnerability, we present DEGNN, a novel GNN model designed to adeptly mitigate noise in both edges and node features. The core idea of DEGNN is to design two separate experts: an edge expert and a node feature expert. These experts utilize self-supervised learning techniques to produce modified edges and node features. Leveraging these modified representations, DEGNN subsequently addresses downstream tasks, ensuring robustness against noise present in both edges and node features of real-world graphs. Notably, the modification process can be trained end-to-end, empowering DEGNN to adjust dynamically and achieves optimal edge and node representations for specific tasks. Comprehensive experiments demonstrate DEGNN's efficacy in managing noise, both in original real-world graphs and in graphs with synthetic noise.