MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks
This work provides curvature-aware training guarantees for neural networks on manifolds, which is significant for researchers and practitioners working with geometric deep learning, especially in fields like surface imaging, physics-informed learning, and robotics on Lie groups.
This paper introduces Manifold Sobolev Informed Neural Optimization (MSINO), a training framework for neural networks on Riemannian manifolds that uses a covariant Sobolev loss and Laplace-Beltrami smoothness regularization. The authors derive geometry-dependent constants that provide a Descent Lemma, a Sobolev Polyak-Lojasiewicz inequality for linear convergence, and a two-step Newton-Sobolev method with local quadratic contraction.
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.