Geometric Invariance of Gradient-Induced Coupling for Continuous 3D Dynamics

We develop an $E(3)$-equivariant graph neural network with continuous latent dynamics, using a shared continuous vector field across nodes and trajectories. Motivated by the regularities shared across diverse physical states, we use the same field across latent states. Because the same field is evaluated at different latent states, gradient updates induce cross-state coupling through the shared-field parameters. We characterize this gradient-induced cross-state coupling and introduce a geometric symmetry bias that makes it invariant under common latent transformations and dependent on relative latent configuration. Our experiments show that invariant coupling can be reflected in latent organization. Across 3D dynamics benchmarks, the continuous formulation remains competitive with established baselines in forecasting performance, even under random irregular temporal sampling. Importantly, the framework unifies continuous latent dynamics with explicit geometric control over cross-state coupling without sacrificing physical $E(3)$ equivariance.