Your Physics Prior Might Be a Constant: Controls for Physics-Informed Learning

Physics-informed machine learning injects a physical law into training and credits the physics when accuracy improves. On a public wildfire-spread benchmark, we show that this attribution can fail, and that two inexpensive checks expose the failure. The first is a pre-training diagnostic: rank the cells the prior is supposed to discriminate and score the ranking against the class prior. The standard fire-spread law (Rothermel) scores 0.210 against 0.193, nearly uninformative at the data's resolution, which correctly predicts that an eikonal physics loss changes nothing. The second is a set of post-training controls. Distilling a calibrated Rothermel teacher into a U-Net gains +0.104 AP over our reproduced Dice baseline on 12 of 12 folds, a result the standard two-arm design would credit to physics. Yet an information-free constant teacher reproduces the gain, a plain Dice+BCE loss reaches +0.095 with no teacher at all, and a genuinely informative learned prior (a 13k-parameter neural cellular automaton) gains less than a constant at the label base rate. The gain comes from a constant-target regulariser whose strength is set by the target level and not by the prior's information. Compared with an information-free control at its own level, the learned prior transmits a small amount of information (+0.03) and the physics prior none. We propose both checks as standard practice for physics-informed claims.