TITLE:
A Unified Geometric and Energetic Framework for Deep Neural Networks via RKHS Embeddings
AUTHORS:
Heriony Rapelanoro-Rabenja
KEYWORDS:
RKHS Geometry, Riemannian Learning, Extrinsic Curvature, Mean Curvature, Geometric Regularization, Energy-Based Models
JOURNAL NAME:
Applied Mathematics,
Vol.17 No.8,
August
28,
2026
ABSTRACT: This article develops a unified geometric and energetic framework for the analysis of deep neural networks, based on embedding the output manifold into a Reproducing Kernel Hilbert Space (RKHS). This embedding induces a natural Riemannian metric, a Levi-Civita connection, a second fundamental form, and a mean curvature vector, allowing the construction of a complete geometric energy model. We show how these tools lead to intrinsic learning dynamics, coherent geometric regularization, and physically interpretable energy flows. Experiments demonstrate improvements in stability, robustness, and generalization.