Conformal Prediction Under Physical Constraints for Scientific Surrogates

Scientific machine learning surrogates replace expensive physics simulators but provide no signal when their predictions cannot be trusted. Existing methods lack coverage guarantees for the structured outputs they produce, and the constraints available at deployment are themselves approximate: a conserved quantity known only to climatological accuracy, or an energy shell fitted from data. We present a physics-constrained conformal prediction framework extending distribution-free coverage guarantees to weather, molecular dynamics, and pharmacokinetic surrogates by projecting conformal sets onto physically valid manifolds. For linear constraints, projection is non-expansive and can only improve coverage; for quadratic constraints, we derive a curvature-corrected ε-relaxed bound for approximately satisfied constraints. An adaptive mechanism widens intervals under distribution shift, with its cap selected only on held-out shifted data to preserve the guarantee. Across three domains, conformal prediction achieves near-nominal coverage where uncalibrated deep ensemble, Monte Carlo dropout, and Bayesian baselines fail outright, and stays 3.9–5.2× tighter in weather and pharmacokinetics once those baselines are conformalized, confirmed for weather by resampling over both training seeds and calibration splits, which shows coverage variance is dominated by the calibration draw (1.95pp) rather than by surrogate training (0.19pp). Selective ε-relaxed projection holds coverage within 2pp of nominal on all four molecular dynamics benchmarks, where exact projection costs up to 17pp, a gap we trace to the curvature of the energy shell rather than to how the energy target is derived. The adaptive detector recovers coverage under weather shift but only partially under molecular and pharmacokinetic shift, reflecting a hard limit rather than an implementation gap: an input-space signal is being asked to answer an epistemic question about the surrogate's error, and cannot always do so.