When Boundary Structure Helps: A Controlled Study of Neural PDE Operators

Boundary conditions provide inputs to neural operators for partial differential equations (PDEs), but can score structure improve learning when information and access are matched? We study the Boundary Heterophily Operator (BHO), a quadratic residual on boundary-key logits. On the Boundary-Embedded Neural Operators (BENO) benchmark with 100 examples used for gradient training, BHO reduces held-out relative-$L_2$ error by 13.87% on the source shape and 8.27% under cross-shape transfer. The low-data advantage persists under two matched training budgets. The full quadratic also outperforms a cross-only score with the same boundary conditioner in BENO and a fixed-source Dirichlet circle problem. A secondary metric computed from the same circle predictions favors BHO over the host, a boundary-group bias and the cross-only score in predicting boundary-induced solution changes. Native mixed-boundary comparisons do not establish BHO's superiority over the host or a scalar message gate, and the tested elasticity comparison favors these controls. In a separate study with constant conditioning and physical-interior queries, removing within-head sign mixing increases error by 23.19% with a shared query/key residual projection but has no detected cost with separate projections. Thus boundary-score parameterization can improve learning within a matched host and route, while sign-restriction costs depend on the tested residual parameterization.