Spectral Degeneracy Operators: A Mathematical Foundation for Physics-Informed Turbulence Modeling

This paper establishes a rigorous mathematical foundation for Spectral Degeneracy Operators (SDOs) and their integration into physics-informed neural networks for turbulence modeling. We introduce a novel class of anisotropic degenerate elliptic operators with separable weight structures that serve dual purposes as analytical tools and trainable neural network components. Our three principal contributions include: (1) A comprehensive fractional regularity theory proving that solutions to La,θu = f gain up to min{s, δ} fractional derivatives in weighted Sobolev-Besov spaces, with explicit dependence on degeneracy exponents θi ∈ [1, 2); (2) A spectral stability theorem for deep SDO networks demonstrating bounded mode amplitude variations of order O(δ log(1/δ)) under parameter perturbations, preventing catastrophic spectral drift during training; (3) A Γ-convergence framework establishing variational limits of discrete SDO energy functionals. These theoretical advances are supported by strengthened proofs for inverse calibration stability and universality of divergence-free SDO closures. Our work bridges degenerate PDE theory with modern machine learning, providing rigorous guarantees for stability, interpretability, and convergence of neural operator architectures in turbulence modeling applications.