Hybrid symbolic regression for data-driven discovery: Governing dimensionless numbers in supercritical heat transfer

With the increasing global demand for high-efficiency and low-emission energy systems, supercritical fluids have gained attention due to their superior thermal properties, thereby posing new challenges for accurate modeling of their complex heat transfer behavior. In this context, interpretable and generalizable models become essential, where scaling analysis helps reduce complexity and reveal governing mechanisms. This study proposes an original framework for automatic construction of dimensionless number systems, inspired by traditional dimensional analysis but extended via modern machine learning techniques. The core innovation lies in a hybrid symbolic regression neural network (HSRNN), which modularizes governing equations and embeds dimensional invariance into its architecture, enabling the generation of physically meaningful and compact base dimensionless numbers. To enhance clarity and robustness, dimensional optimization and expression refinement are performed. Using supercritical heat transfer as a case study, this work analyzes 1492 experimental data points under seven operating conditions. The base dimensionless groups are further interpreted using classical dimensional analysis and reduced via the active subspaces method, identifying key factors related to mass, momentum and energy conservation. The proposed framework integrates the strengths of physical modeling, symbolic regression, and deep learning, and is validated through a representative case of supercritical heat transfer, highlighting its applicability and potential for modeling complex physical systems.