  Graph Neural Networks (GNNs) with equivariant properties have emerged as
powerful tools for modeling complex dynamics of multi-object physical systems.
However, their generalization ability is limited by the inadequate
consideration of physical inductive biases: (1) Existing studies overlook the
continuity of transitions among system states, opting to employ several
discrete transformation layers to learn the direct mapping between two adjacent
states; (2) Most models only account for first-order velocity information,
despite the fact that many physical systems are governed by second-order motion
laws. To incorporate these inductive biases, we propose the Second-order
Equivariant Graph Neural Ordinary Differential Equation (SEGNO). Specifically,
we show how the second-order continuity can be incorporated into GNNs while
maintaining the equivariant property. Furthermore, we offer theoretical
insights into SEGNO, highlighting that it can learn a unique trajectory between
adjacent states, which is crucial for model generalization. Additionally, we
prove that the discrepancy between this learned trajectory of SEGNO and the
true trajectory is bounded. Extensive experiments on complex dynamical systems
including molecular dynamics and motion capture demonstrate that our model
yields a significant improvement over the state-of-the-art baselines.
