From data chaos to physically interpretable deterministic mapping.

Discovering governing equations directly from observational data remains a fundamental challenge in science and engineering, particularly when measurements are noisy, high-dimensional, or multi-scale. Existing approaches often cast equation discovery as a regression problem that selects candidate terms to fit observed trajectories, which can limit structural stability and identifiability under realistic data conditions. We propose a structured operator-learning framework that reformulates equation discovery as a constrained dynamical inference problem integrating spectral decomposition, physics-guided sparse projection, and cross-view consistency regularization within a unified architecture. By decomposing dynamics into scale-resolved components and enforcing invariance across perturbed observations, the framework promotes stable and interpretable equation recovery. Here, we show that the method consistently identifies compact governing equations while maintaining strong long-horizon predictive accuracy across canonical nonlinear systems and representative industrial processes, even under noisy and distribution-shifted data.