The classical dimensional analysis provides powerful insights into underlying physical mechanisms, but has limitations in determining the uniqueness and measuring the relative importance of dimensionless quantities. To address these limitations, we propose a data-driven approach, called mutual-information-based dimensional learning, to identify unique and relevant dimensionless quantities from available data. The proposed method employs a novel information-theoretic criterion to measure the relative importance of dimensionless quantities, whereas the existing methodologies rely on sensitivity/derivative-based measures. This entropy-based measure provides two significant advantages: (1) invariance (objectivity) with respect to reparametrizations of variables, and (2) robustness against outliers. Numerical results show that our method outperforms the current state-of-the-art method in these aspects, and enables identifying dominant dimensionless quantities. Examples include the study of the friction factor in benchmark pipe flows, the eddy viscosity coefficients in turbulent channel flows and the vapor depression dynamics in laser–metal interaction.

中文翻译：

---

基于互信息量纲的量纲学习：用于识别相关无量纲量的客观算法


经典的量纲分析为潜在的物理机制提供了强大的见解，但在确定唯一性和测量无量纲量的相对重要性方面存在局限性。为了解决这些限制，我们提出了一种数据驱动的方法，称为基于互信息的数据量纲学习，以从可用数据中识别唯一且相关的无量纲量。所提出的方法采用一种新的信息论标准来测量无量纲量的相对重要性，而现有的方法依赖于基于敏感性/导数的测量。这种基于熵的测量提供了两个显著的优势：（1） 变量重新参数化的不变性（客观性），以及 （2） 对异常值的鲁棒性。数值结果表明，我们的方法在这些方面优于当前最先进的方法，并且能够识别主要的无量纲量。示例包括对基准管道流中的摩擦系数、湍流通道流中的涡流粘度系数以及激光-金属相互作用中的蒸汽抑制动力学的研究。