Test-Time Compute Allocation in Random-Feature PDE Solvers: Width versus Search

Under a fixed work budget, should a random-feature partial differential equation (PDE) solver widen one trial space or search across several? We compare these allocations using the same feature distribution and fitting procedure. An exact error decomposition separates candidate-set quality from final selection loss using a post-solve reference-error oracle. Across 60 independently sampled linear advection–diffusion–reaction problems that activate search, six budgets, and four seeds, the task-geometric search/wide error ratio is 9.32 (95% CI 7.92–10.91); search wins on none of the 60 tasks under this paired statistic. Oracle selection from the same candidate sets still yields a ratio of 9.26, while final selection adds only 0.7% to the geometric mean error. Thus, almost the entire gap is already present in the available candidates. Ablations nevertheless show that physics-based selection is useful within those populations. Finite-set ranking guarantees and a spectral residual bound for a restricted elliptic operator class clarify when verification controls ranking and solution error. These findings identify candidate generation and compute allocation as the main bottlenecks for the tested search policy: accurate final verification cannot compensate for the absence of a competitive candidate.