Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transport

Adaptive meshes enable neural operators for partial differential equations (PDEs) to allocate spatial samples and computational resources according to local physical structures. Existing approaches, however, primarily focus on where to compute, while paying less attention to how to interact after node relocation. Mesh adaptation changes local sampling scales, neighborhood structures, and geometric contexts, making representations formed at different nodes not necessarily directly comparable. Direct aggregation may therefore entangle genuine physical variation with discretization-induced representation variation. Moreover, because allocation and interaction are jointly optimized through the same output objective, their individual roles are difficult to distinguish from final errors alone. We introduce operator readability, which requires an adaptive operator to explicitly account for and test why computation is allocated to particular locations and how representations interact under the resulting nonuniform discretization. Based on this principle, we propose the Gauge-Aware Adaptive Mesh Neural Operator (GA-AMNO). Physics-informed adaptive allocation answers where to compute, while geometry-conditioned low-rank Gauge transport maps source features into target representation contexts before aggregation, answering how to interact. This design turns the otherwise implicit mesh-to-solver information exchange into an inspectable and intervenable computational process. We further establish sufficient conditions for representation-consistent aggregation and analyze approximate transport errors and continuity under topology-preserving mesh deformations. Experiments across five PDE benchmarks demonstrate improved predictive accuracy, while controlled interventions and geometric-mismatch analyses verify the computational roles of allocation and interaction and show that Gauge transport improves cross-discretization representation compatibility under strong geometric mismatch.