Partition-of-Unity Adapters for Efficient Local Refinement of Implicit PDE Representations

Pretrained implicit PDE surrogates and neural operators often produce imperfect predictions on new problem instances, while dense interior target solutions are unavailable for adaptation. Full fine-tuning updates the entire model, and weight-space adapters reduce trainable parameters but do not explicitly localize corrections in the physical domain. We introduce Partition-of-Unity Adapters, a coordinate-space adaptation layer for frozen PDE representations. The method attaches smooth overlapping windows to the domain and equips each window with a low-dimensional Chebyshev correction packet, so every trainable coefficient is tied to a known spatial support and local basis mode. Residual or physical- change signals select local correction packets, with coordinated relaxation or dense activation available for broader or coupled responses. The adapter is fit from PDE residuals, initial and boundary conditions, interface constraints, and optional sparse measurements, without dense interior solution labels during deployment adaptation. Across frozen Fourier-feature MLPs, FNO backbones, localized numerical repair problems, and material and interface changes with nonlocal responses, POU adapters often improve predictions over frozen models and matched global coordinate corrections. Compared with full fine-tuning, they use substantially fewer trainable parameters and shorter adaptation times under the evaluated optimization budgets. Selected local corrections leave the frozen field unchanged outside their supports by construction. The results also identify clear scope boundaries: transport-dominated errors, singularities, and strongly coupled responses can require PDE- aware supports or bases, coordinated relaxation, dense capacity, or full model adaptation.