AI and the World's Hard Problems · Theoretical computer science · arXiv · 2026-10-21
Learning Normal Coordinates on Riemannian Manifolds with Flow Charts
Riemannian manifolds are powerful models for problems with non-Euclidean structure, but the repeated evaluation of exponential and logarithmic maps required by geometric algorithms is computationally costly. The authors train a conditional invertible neural network, called a flow chart, to approximate normal coordinates around points on a given manifold, so both maps reduce to fast Euclidean operations in the learned coordinate space, supported by an efficient nonnegative loss that is zero precisely when the chart is normal. Across sixteen synthetic and data-derived geometries, pretrained flow charts deliver a median 60-70x speedup over numerical baselines at comparable logarithmic map error, and their learned coordinates improve the accuracy of a numerical geodesic solver.