Physics-constrained symbolic regression for reduced-order carinal pressure laws in transient endotracheal-tube airflow

High-fidelity computational fluid dynamics (CFD) can resolve transient airway flow during mechanical ventilation, but its cost limits parametric exploration and rapid model evaluation. We present a physics-constrained symbolic regression framework that converts transient three-dimensional multiphysics CFD simulations of endotracheal-tube airflow into closed-form reduced-order laws for local carinal pressure. The database spans pressure-controlled ventilation (PCV), volume-controlled ventilation (VCV), oxygen mass fraction, tip-to-carina distance, Murphy-eye orientation, and the transient inlet-flow response. All regressors are expressed as dimensionless variables, so that the search is unit-invariant, and nonlinear operators act only on dimensionally admissible arguments. Separate symbolic searches are performed for pressure-controlled and VCV. The recovered laws are consistent with mode-specific ventilation physics: an inverse quadratic inspiratory phase dependence for PCV and a normalized-time Reynolds number dependence for VCV. The best expressions achieve a coefficient of determination of 0.9993 and a root mean square error (RMSE) of 6.98×10−4 for PCV, and a coefficient of determination of 0.9983 and a root mean square error of 1.75×10−3 for VCV, reducing the prediction error by approximately one order of magnitude relative to a multivariate-linear baseline. Multi-seed Pareto screening and dimensionless pressure-regime maps show that the surrogate enables rapid interpolation within the sampled parameter space without rerunning CFD. The framework provides a transparent, scientific machine learning route from multiphysics flow simulations to dimensionally admissible pressure laws for biological fluid mechanics, while remaining limited to the investigated airway geometry and operating conditions.