Stratified Conformal Prediction for Neural Fluid Surrogates: Spatially Adaptive Uncertainty Quantification Framework

Abstract

Deep learning surrogates, especially neural operators, provide rapid alternatives to traditional CFD solvers; nonetheless, they lack inherent physical guarantees, a significant drawback for safety-critical engineering applications. Conformal prediction offers distribution-free coverage guarantees; yet, traditional global calibration is inadequate for multi-regime fluid systems: a single threshold is overly conservative in laminar areas and insufficient in turbulent sections. We propose a physics-informed, regime-stratified conformal prediction approach for neural fluid surrogates that addresses the gap in three phases. First, we formulate a dimensionally consistent physics residual error based on the steady-state incompressible Navier-Stokes momentum equations, which serves as an unsupervised nonconformity score, validated by strong Spearman rank correlation with the actual prediction error (0.94 for cavity, 0.90 for cylinder). Second, we use regime-stratified calibration, dividing the parametric space according to Reynolds number and boundary-condition type to produce regime-specific thresholds that are up to 99.69\% tighter than the global bound, while ensuring around 90\% coverage across all regimes. Third, a modular spatially adaptive scaling stage converts uniform per-regime bounds into pixel-level uncertainty maps using any spatially varying signal; three candidates, vorticity magnitude, velocity magnitude and MC dropout standard deviation, are systematically evaluated with independently optimised hyperparameters. Against standalone MC dropout, the proposed framework produces intervals that are 3.3–16.2 times narrower at the same coverage target, with higher spatial error correlation. Among spatial signals within the framework, vorticity magnitude is adopted as the default: it is the only candidate that improves over the flat baseline across all regimes on both benchmarks at negligible computational cost relative to the 50-pass MC dropout alternative. A pressure-gradient extension demonstrates framework modularity across field variables, indicating that pressure prediction errors are more spatially concentrated than velocity errors, accounting for an additional efficiency improvement. All results are validated by a five-component statistical robustness suite, which includes bootstrap resampling, K-fold cross-validation, multi-seed stability, significance-level sweep, and calibration-size ablation. The framework is demonstrated on lid-driven cavity and cylinder flow benchmarks with a Fourier neural operator surrogate.