Spectral Degeneracy Operators for Interpretable Turbulence Modeling

Spectral Degeneracy Operators (SDOs) provide a unified framework for embedding physical symmetries and adaptive singularities into neural network layers. In this work we develop the theoretical and computational underpinnings of SDOs and demonstrate their role in neural symmetrization and turbulence modeling. We establish a generalized spectral decomposition for vector-valued SDOs, derive Lipschitz stability estimates for the inverse calibration of degeneracy points from sparse or boundary data, and prove a neural–turbulence correspondence theorem showing that SDO-based networks can approximate turbulence closure operators while preserving incompressibility. This combination of rigorous spectral theory, inverse problem analysis, and physics-informed neural design bridges harmonic analysis, degenerate PDEs, and fluid dynamics. Our results provide both mathematical guarantees and practical algorithms for constructing data-driven yet physically consistent turbulence models. By enforcing modewise stability under training and exploiting Green’s function representations, SDO layers act as adaptive spectral filters that learn anisotropic structures without violating conservation laws. The proposed framework opens new directions for interpretable deep learning architectures grounded in the spectral properties of underlying physical operators.