Symmetry Descent Neural Operators: From Local Equivariance to Global Consistency

Equivariant neural networks commonly assume that a group action and its representations are known and consistent over the entire domain. Heterogeneous materials, local defects, and domains with holes make symmetry valid only locally; the relations between local frames vary across system instances, and loops can also carry nontrivial topological information. We propose Symmetry Descent Neural Operators (SDNO), which treat local symmetries and frame alignment as episode-conditioned structures determined from operator observations. Given multiple input–output observations from the same operator, SDNO learns the feature transformations that can be shared within each local region and the feature alignment between neighboring regions. These constraints define a cross-region weight-sharing space, while curvature and holonomy represent local defects and global topology, respectively. A differentiable projection parameterizes this space as trainable neural operator layers. Our theoretical results characterize when local equivariant layers compose into a global operator and establish stability and identifiability under structure-estimation error. The experiments cover local transport with the unit complex phase group $\mathrm{U}(1)$ and the noncommutative special unitary group $\mathrm{SU}(2)$, operator prediction and topology transfer under hidden local transport, polycrystalline local frames, and a public Structure-from-Motion rotation averaging benchmark. SDNO maintains accurate structure recovery and operator prediction under unseen changes in holonomy, curvature, topology, and local frames, and improves global orientation estimation on public noisy relative-rotation data.