Spectral Degeneracy Operators: Bridging Physics-Informed Machine Learning and Degenerate PDEs

This work establishes a comprehensive mathematical theory for Spectral Degeneracy Operators (SDOs), a novel class of degenerate elliptic operators that encode physical symmetries and adaptive singularities through principled degeneracy structures. We develop the fundamental analytic framework, proving generalized spectral decompositions, Weyl-type asymptotics with explicit Bessel function connections, and maximum principles for vector-valued degenerate systems. The theory extends to non-Euclidean domains, with Landau-type inequalities establishing sharp uncertainty principles between spatial and spectral localization. For neural applications, we introduce SDO-Nets architectures with mathematically guaranteed well-posedness, stability, and physical consistency and prove a neural-turbulence correspondence theorem connecting learned parameters to underlying turbulent structures. Inverse problem analysis provides Lipschitz stability for degeneracy point calibration from sparse data. This work bridges degenerate PDE theory, harmonic analysis, and physics-informed machine learning, providing rigorous foundations for data-driven yet physically consistent modeling of complex systems.