标题： 分层无量纲学习（Hi-π）：一种物理数据混合驱动的方法，用于发现无量纲参数组合 显示英文标题

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

摘要： 量纲分析为减少物理复杂性和揭示固有定律提供了通用框架。 然而，其在高维系统中的应用仍然会产生冗余的无量纲参数，使得建立具有物理意义的描述变得困难。 在此，我们引入分层无量纲学习（Hi-{\pi }），一种结合量纲分析和符号回归的物理数据混合驱动方法，可自动发现关键的无量纲参数组合。 我们将该方法应用于流体力学各个研究领域的经典例子。 对于雷诺-贝纳德对流，该方法准确提取了两个内在无量纲参数：雷诺数和普朗特数，验证了其在多尺度数据中的统一表示优势。 对于圆管中的粘性流动，该方法自动发现了两个最优无量纲参数：雷诺数和相对粗糙度，实现了准确性和复杂性之间的平衡。 对于亚音速流动中的可压缩性修正，该方法有效提取了经典的可压缩性修正公式，同时通过最优参数变换展示了其发现分层结构表达式的能力。 显示英文摘要

摘要： 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.