StarVortex: Self-Supervised Geometric Representation Learning of Collective Integral Curves for Lagrangian Vortices

Learning useful representations of dynamical systems from geometry alone is challenging: individual trajectories provide limited information, while the relevant structure often lies in the collective deformation of neighboring trajectories and should be insensitive to arbitrary spatial orientation. We study this problem in fluid flows and introduce StarVortex, a self-supervised framework for learning geometric representations directly from local bundles of integral curves. We represent each neighborhood as a star consisting of a center curve and symmetrically seeded neighbors, exposing both center-curve transport and relative bundle deformation. Rather than requiring a fully rotation-equivariant architecture, StarVortex combines an exact architectural bias induced by the discrete symmetry of the sampling stencil with a representation-level objective that enforces consistency under arbitrary continuous rotations. Reconstruction and variance–covariance regularization further encourage the latent space to retain informative geometric variation. The resulting representation captures vortex-relevant structure without velocity, vorticity, physics-derived descriptors, or training labels. It supports label-free discovery of vortex regions and provides transferable features for supervised recognition of vortex regions, corelines, and hairpin vortices across pathlines, streamlines, and vortex lines. Experiments and ablations show that explicitly structuring self-supervision around geometric symmetry is critical for learning useful representations, and that the learned features are particularly beneficial when labels are scarce or evaluation shifts beyond the training domain.