  Learning to reason about relations and dynamics over multiple interacting
objects is a challenging topic in machine learning. The challenges mainly stem
from that the interacting systems are exponentially-compositional, symmetrical,
and commonly geometrically-constrained. Current methods, particularly the ones
based on equivariant Graph Neural Networks (GNNs), have targeted on the first
two challenges but remain immature for constrained systems. In this paper, we
propose Graph Mechanics Network (GMN) which is combinatorially efficient,
equivariant and constraint-aware. The core of GMN is that it represents, by
generalized coordinates, the forward kinematics information (positions and
velocities) of a structural object. In this manner, the geometrical constraints
are implicitly and naturally encoded in the forward kinematics. Moreover, to
allow equivariant message passing in GMN, we have developed a general form of
orthogonality-equivariant functions, given that the dynamics of constrained
systems are more complicated than the unconstrained counterparts.
Theoretically, the proposed equivariant formulation is proved to be universally
expressive under certain conditions. Extensive experiments support the
advantages of GMN compared to the state-of-the-art GNNs in terms of prediction
accuracy, constraint satisfaction and data efficiency on the simulated systems
consisting of particles, sticks and hinges, as well as two real-world datasets
for molecular dynamics prediction and human motion capture.
