When Does Structure Help? Inductive Bias under Structural and Semantic Distribution Shift in Learned Control

Learned controllers for structured dynamical systems must often generalize after deployment, when either the system's interaction structure changes or the processes acting on that structure change. These shifts are distinct, yet they are usually evaluated together. As a result, it is unclear whether an architectural inductive bias improves out-of-distribution (OOD) control because it matches the source of the shift, or merely because it matches the visible structure of the system. We separate OOD generalization into two axes: structural shift, which changes system topology, and semantic shift, which changes exogenous workload distributions while preserving topology. Crossing them yields four controlled settings: in-distribution, structural OOD, semantic OOD, and combined OOD. In a physics-grounded networked-control testbed, we compare three policy classes with different structural assumptions: a Transformer with fixed component slots, the same Transformer with slot identity removed to enforce permutation invariance, and a graph neural controller that explicitly encodes the physical interaction graph. All policies are trained under a common PPO protocol and evaluated over 20 training seeds, five held-out topologies, and measured AI-workload traces used to induce semantic shift. In a weakly constraint-active regime, explicit graph structure provides no statistically supported advantage over permutation invariance under structural OOD, while domain randomization does not reliably improve generalization to the measured semantic shift. These results show that the presence of a physical graph alone is not sufficient to establish a benefit from graph inductive bias in OOD control. To test whether its value depends on the graph becoming decision-relevant, we introduce a pre-specified, model-independent physics calibration procedure that constructs a feasible constraint-active regime without using learned-policy outputs or screening the final held-out OOD panel. This creates a controlled setting in which the role of structural inductive bias can be tested without post-hoc benchmark selection. Our contribution is a factorized framework for diagnosing structural and semantic OOD in learned control, together with a controlled methodology for testing when structural inductive biases become useful.