Relaxing Continuous Constraints of Equivariant Graph Neural Networks for Physical Dynamics Learning
arxiv(2024)
摘要
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.
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