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.
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