Adaptive Symmetry Discovery for Dynamical System Identification

Dynamical systems model trajectory data generated by an underlying fixed dynamics, with applications ranging from biological systems to flows in physics. The identification problem concerns recovering the parameters of a system from observed trajectories. In many scientific settings, however, dynamical systems are not generic and instead exhibit symmetries imposed by physical laws, formalized as equivariance with respect to a group action. In this work, we study \emph{adaptive symmetry discovery} for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.