AdaSolver: Efficient Neural Operator Inference via Test-Time Adaptive Sampling

Neural operators provide efficient surrogates for partial differential equations (PDEs), yet their inference performance is typically tied to uniform spatial discretizations or sampling schemes. Many PDE solutions exhibit localized structures that require dense sampling for accurate prediction, while smooth regions can be represented with substantially fewer points. Naively changing the sampling density, however, shifts the empirical input measure and can introduce sampling-dependent bias into the learned neural operator. We introduce **AdaSolver**, a measure-aware framework for adaptive spatial computation at test time. We characterize sampling-induced bias and its propagation through physics-attention, and derive a sampling-density-consistent Transolver using self-normalized importance weighting. The resulting measure-consistent formulation preserves consistency with the reference measure, enabling adaptive test-time sampling across different physical systems and geometric designs. Building on this formulation, we explore three refinement strategies based on physics-informed priors, coarse-prediction guidance, and cache-aware learned refinement. Across five standard PDE benchmarks and three industrial aerodynamic tasks, AdaSolver improves prediction accuracy under matched sampling budgets and accelerates surrogate-based airfoil and wing optimization. These results establish adaptive spatial sampling as a principled axis of test-time scaling for neural operators.