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

Abstract

Adaptive meshes enable neural operators for partial differential equations (PDEs) to allocate spatial samples and computation according to local physical structures. Existing approaches, however, mainly address where to compute, with less attention to how information should interact after node relocation. Mesh adaptation changes local sampling scales, neighborhood structures, and geometric contexts, so representations formed at different nodes may not be directly comparable. Direct aggregation can therefore entangle physical variation with discretization-induced representation variation. Because allocation and interaction are jointly optimized through the same output objective, their individual roles are also difficult to distinguish from final errors alone. We introduce operator readability, requiring an adaptive operator to 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 makes mesh-to-solver information exchange inspectable and intervenable. We establish sufficient conditions for representation-consistent aggregation and analyze approximate transport errors and continuity under topology-preserving mesh deformations. Experiments on five PDE benchmarks demonstrate improved predictive accuracy, while controlled interventions and geometric-mismatch analyses verify the roles of allocation and interaction and show that Gauge transport improves cross-discretization representation compatibility under strong geometric mismatch.