Dynamical Systems

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 $\textit{geometric graphs}$, $\textit{i.e.}$, graphs with nodes positioned in the Euclidean space given an $\textit{arbitrarily}$ 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 $\textit{Galilean invariance}$. 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.

We show how a pair of points can uniquely represent a left-infinite sequence obtained from observations of an underlying dynamical system through a phenomenon called causal embedding. A driven dynamical system creates such pairs, and a function can be learnt on them that can reconstruct the underlying dynamics as in Takens delay embedding. The approach assures embedding stability unlike Takens delay embedding, and learnability which can be absent while stability can be present in the current reservoir computing framework. This accurately models underlying systems where recent methods like the next-generation reservoir computing fail. We demonstrate results and compare with the other methods including SINDY-PI.