Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.
Figures & tables
Figure 1 : In (a), objects positioned in an arbitrary global 2D coordinate frame; arrows represent orientations. In (b)-(e), objects in the canonicalized local coordinate frames, translated to match the target object’s position and rotated to match its orientation
Figure 2 : Results on synthetic dataset
Method
NRI
dNRI
LoCS
F1
26.5
60.8
88.9
Table 1 : Relation prediction F1 score on synthetic dataset
Figure 3 : LoCS predictions on (a) synthetic dataset, (b) charged particles and (c) inD.
Figure 5 : Total error curves in ablation experiments: (a) on highly interactive charged particles, (b) on the impact of speed normalization, (c) on the impact of isotropic filters, (d) on the impact of rotation.
Figure 6 : In 6(a) , 6(b) , 6(c) , translated-only local coordinate frames for object A in 3 different dynamical systems #1-#3. In 6(d) , dynamical system #3, A’s roto-translated local coordinate frame