Geometric Learning Dynamics

ArXi:2504.14728v3 Announce Type: replace We present a unified geometric framework for modeling learning dynamics in physical, biological, and machine learning systems. The theory reveals three fundamental regimes, each emerging from the power-law relationship $g \propto \kappa^\alpha$ between the metric tensor $g$ in the space of trainable variables and the noise covariance matrix $\kappa$. The quantum regime corresponds to $\alpha = 1$ and describes Schr\"odinger-like dynamics that emerges from a discrete shift symmetry.