Operator-Budget Evaluation of Residual-Adaptive Sampling in Physics-Informed Neural Networks

Residual-adaptive collocation is commonly compared at equal optimizer steps, although maintaining its sampling proposal requires PDE-residual evaluations in addition to those used for training. Such comparisons can obscure whether an apparent gain comes from point placement or from additional information work. We introduce an operator-budget evaluation framework that records training and proposal-maintenance residual calls in a common ledger, distinguishes fixed-update cost from work-matched accuracy, and requires a joint cost–accuracy decision. Amortized residual proposal maintenance (ARPM) serves as a finite-pool case study: it reuses minibatch residuals to repair a persistent proposal cache between periodic global refreshes. In a prospectively specified multi-PDE evaluation, the configuration met the joint primary criterion after maintenance work was charged. Post-review analyses separated writeback from uniform mixing, added a heat-equation family, varied the L-BFGS refinement points and measured implementation overhead. They did not establish a general accuracy advantage of writeback or consistent gains beyond the selected suite. The framework’s practical contribution is an auditable comparison that exposes both useful cost–accuracy trade-offs and competitive simpler schedules. Operator savings must still be assessed alongside elapsed time, memory and problem-specific accuracy requirements.