融合物理与数据，自动发现关键无量纲参数组合
Hierarchical Dimensionless Learning (Hi-π): A physics-data hybrid-driven approach for discovering dimensionless parameter combinations
- 结合量纲分析与符号回归，自动生成无量纲参数
- 在对流、管道流等场景中准确提取经典无量纲数
- 适合流体力学、多尺度建模等领域的研究人员
量纲分析为降低物理系统复杂性、揭示内在规律提供了通用框架。然而，其在高维系统中仍会产生冗余的无量纲参数，难以建立具有物理意义的描述。本文提出分层无量纲学习（Hi-π），一种融合物理与数据驱动的方法，结合量纲分析与符号回归，自动发现关键的无量纲参数组合。应用于流体力学多个经典问题：在瑞利-贝纳德对流中，准确提取出雷利数和普朗特数两个本征无量纲参数，验证了其跨多尺度数据的统一表征优势；在圆管粘性流动中，自动发现雷诺数与相对粗糙度两个最优无量纲参数，兼顾精度与复杂度平衡；在亚音速流压缩性修正中，有效复现经典修正公式，并展示通过最优参数变换发现层级结构表达的能力。
原文摘要 · Abstract (English)
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-π), 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-Bénard 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.
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