Geometry-Aware Operator Families for Structured Representation Learning

The geometry of latent representations governs which components should interact and how information should propagate, making geometry-aware operator design a fundamental ingredient of structured deep representation learning. However, existing neural architectures typically rely on generic operator templates or geometry-specific constructions, creating a need for a unified framework that can derive admissible operators directly from fixed structural information while remaining adaptive to changing contexts. We introduce Geometry-Induced Operator Families (GIOF), a general framework that converts fixed geometry into a structured family of propagation operators and dynamically selects an appropriate member of this family according to the current context. GIOF first transforms geometry-derived interaction channels into reusable generator bases, then combines them through a context-dependent selector and adaptive propagation scale, and finally realizes the selected operator through stable continuous-time propagation and a bottleneck residual layer. We establish theoretical guarantees covering parameter compression, identifiability, stability, locality, compositional structure, and oversmoothing behavior, while controlled experiments validate these mechanisms and experiments on PEMS-BAY and METR-LA achieve the lowest mean MAE across all reported regional-outage settings, improving over the strongest retained baseline by 2.4%--8.8% at 30% missing sensors.