PHYSICS-QUALIFIED SAFEGUARDS FOR RELIABLE ITERATIVE NEURAL OPERATORS

Iterative neural operators can reach a learned fixed point that remains physically invalid, and fast fixed-point acceleration can exacerbate this failure under distribution shift. We introduce PHYSSAFE-IRNO, a reliability wrapper that aligns learned stationarity with physical validity during training and physics-qualifies accelerated proposal at inference. During training, stationarity-conditioned physical-risk supervision discourages physically invalid learned fixed points, while averagedness-oriented regularization shapes the refinement map for safeguarded fixed-point updates. At inference, a physics-qualified SuperMann solver screens numerically admissible fast proposals using a learned–physical bi-residual filter and falls back to separating-halfspace or Krasnosel’skii–Mann updates when needed. Under the explicit assumption that the learned refinement map is exactly averaged, the wrapper inherits SuperMann convergence, under a calibrated residual-envelope condition, sufficiently small learned residual implies finite-tolerance physical validity. Prediction-only temporal residuals avoid dependence on future ground-truth states, and a disjoint physical audit evaluates returned states using separately implemented diagnostics. Across four PDE systems and six held-out shifts, PHYSSAFE-IRNO reduces aggregate error from 0.087 for physics-aware Anderson acceleration and 0.092 for SuperMann to 0.068, cuts the false-fixed-point rate to 1.3%, and raises joint success to 96.8%. The gains over fixed IRNO transfer across FNO, TFNO and CNO at 1.16× the wall-clock cost of fixed-step IRNO. The implementation code is provided in the Supplementary Material.