Hierarchical Dimensionless Learning (Hi-π): A physics-data hybrid-driven approach for discovering dimensionless parameter combinations

Abstract: Dimensional analysis provides a universal framework for reducing physical complexity and reveal inherent laws. However, its application to high-dimensional systems still generates redundant dimensionless parameters, making it challenging to establish physically meaningful descriptions. Here, we introduce Hierarchical Dimensionless Learning (Hi-{\pi}), a physics-data hybrid-driven method that combines dimensional analysis and symbolic regression to automatically discover key dimensionless parameter combination(s). We applied this method to classic examples in various research fields of fluid mechanics. For the Rayleigh-B\'enard convection, this method accurately extracted two intrinsic dimensionless parameters: the Rayleigh number and the Prandtl number, validating its unified representation advantage across multiscale data. For the viscous flows in a circular pipe, the method automatically discovers two optimal dimensionless parameters: the Reynolds number and relative roughness, achieving a balance between accuracy and complexity. For the compressibility correction in subsonic flow, the method effectively extracts the classic compressibility correction formulation, while demonstrating its capability to discover hierarchical structural expressions through optimal parameter transformations.