Relaxing Continuous Constraints of Equivariant Graph Neural Networks for Physical Dynamics Learning

TL;DR

This paper introduces DEGNN, a graph neural network that effectively models physical dynamics with discrete symmetries by relaxing continuous equivariance constraints, leading to improved accuracy and generalization.

Contribution

DEGNN is a novel discrete equivariant GNN that guarantees symmetry to discrete groups and relaxes continuous constraints for better physical dynamics modeling.

Findings

DEGNN outperforms state-of-the-art methods in 20 physical scenarios.

It is more data-efficient, requiring less training data.

DEGNN generalizes well to unobserved orientations.

Abstract

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.