Learning When to Trust PDE Coupling Priors in Neural Operators

Coupled neural operators must decide how strongly to use interactions suggested by the governing equations. We study this structural decision under finite data using controlled interventions that vary cross-field coupling while holding the operator backbone and evaluation fixed. On Active Matter and a turbulent radiative layer, fixed PDE coupling priors exhibit different data-scaling behavior, and leave-one-pathway-out interventions show that the predictive effect of individual interactions can change sharply with the data regime. Motivated by this observation, we factor each cross-field interaction into a learned operator and a scalar pathway strength. A short warmup provides pathway-specific evidence to initialize these strengths before joint training. On scarce Active Matter data, the resulting model reduces relative error from $7.06\%$ for dense coupling and $6.77\%$ for the fixed PDE graph to $6.44\%$. Across Active Matter and the turbulent radiative layer, the same method improves on the strongest fixed structural baseline in all four scarce and medium regimes and remains within 0.08 percentage points of the best fixed choice in both abundant regimes. With the method frozen before evaluation on Rayleigh–B\'enard convection, it improves scarce-data error from $0.0872$ to $0.0831\pm 0.0009$ and matches the strongest fixed baseline with abundant data. These results support calibrating PDE coupling structure at the level of individual pathways using target-regime evidence.