Dubins-Aware NCO: Learning SE(2)-Equivariant Representations for Heading-Constrained UAV Routing

The nonholonomic constraints of fixed-wing UAVs, characterized by coupled heading-curvature feasibility and asymmetric costs, fundamentally deviate from classical Euclidean routing assumptions. While standard neural combinatorial optimization (NCO) architectures could theoretically incorporate Dubins costs via reward signals, such naive adaptations lack the capacity to explicitly model the intrinsic SE(2) geometric invariance and directional asymmetry of fixed-wing motion, leading to suboptimal generalization. To bridge this gap, we propose a Dubins-Aware NCO framework. We design a dual-channel embedding to decouple asymmetric physical distances from rotation-stable geometric features. Furthermore, we introduce a Rotary Phase Encoding (RoPhE) mechanism that theoretically guarantees strict SO(2) equivariance within the attention layer. Extensive sensitivity, ablation, and cross-distribution generalization experiments are conducted on tasks spanning varying turning radii and problem variants with instance scales of 10, 20, 36, and 52 nodes. The results consistently validate the superior optimality and stability of our approach compared with state-of-the-art DRL and NCO baselines, while maintaining significant computational efficiency advantages over classical heuristics. Our results highlight the importance of explicitly embedding geometry-physics consistency, rather than relying on scalar reward signals, for real-world fixed-wing autonomous scheduling.