Symmetry Descent Neural Operators: From Local Equivariance to Global Consistency

Many physical operators are locally symmetric yet globally heterogeneous: frames vary, defects create curvature, and loops carry holonomy. We introduce Symmetry Descent Neural Operators (SDNO), which infer this operator-specific structure from a few input--output observations. SDNO recovers local Lie subspaces, aligns them by learned edge transport, and uses a differentiable soft constraint map to regularize local and cross-patch compatibility residuals. For the flat descent component with finite-dimensional fibers and unitary transport, we characterize globally compatible parameters through loop holonomy and derive stability controls for soft projection, approximate compatibility, and local subspace recovery under an excitation gap. Controlled $\mathrm U(1)$, magnetic Schr\"odinger, and polycrystalline studies show recovery of hidden connection and frame structure and improved held-out prediction under changing connections and material frames.