MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks

We introduce Manifold Sobolev Informed Neural Optimization (MSINO), a curvature aware training framework for neural networks defined on Riemannian manifolds. The method replaces standard Euclidean derivative supervision with a covariant Sobolev loss that aligns gradients using parallel transport and improves stability via a Laplace Beltrami smoothness regularization term. Building on classical results in Riemannian optimization and Sobolev theory on manifolds, we derive geometry dependent constants that yield (i) a Descent Lemma with a manifold Sobolev smoothness constant, (ii) a Sobolev Polyak Lojasiewicz inequality giving linear convergence guarantees for Riemannian gradient descent and stochastic gradient descent under explicit step size bounds, and (iii) a two step Newton Sobolev method with local quadratic contraction in curvature controlled neighborhoods. Unlike prior Sobolev training in Euclidean space, MSINO provides training time guarantees that explicitly track curvature and transported Jacobians. Applications include surface imaging, physics informed learning settings, and robotics on Lie groups such as SO(3) and SE(3). The framework unifies value and gradient based learning with curvature aware convergence guarantees for neural training on manifolds.