Metric Gradient Flows and Curvature-Aware Learning in Functional Model Spaces

We recast learning dynamics on the quotient model space F as gradient flows in a metric space, following the Ambrosio–Gigli–Savaré (AGS) framework. This replaces parameter-space gradient descent with an intrinsic, architecture-independent flow on function classes. Under geodesic F-convexity of the loss functional we obtain existence, uniqueness, and exponential contraction of learning flows, and every local minimizer is global. Where F carries additional structure (an Alexandrov curvature bound, or locally a smooth Riemannian metric), Jacobi-field and comparison arguments quantify the sensitivity of learning trajectories to perturbations of data or loss. Curvature-based stability statements are given as comparison theorems with explicit hypotheses, or clearly marked as heuristics.