用神经网络自动发现数据中的相似性关系，揭示物理定律。
Similarity Learning with neural networks
- 通过神经网络学习数据中的相似性，逼近无量纲量的物理规律。
- 在层流、非牛顿流及湍流等复杂流动中验证了方法的有效性。
- 适合对物理规律挖掘感兴趣的科研人员使用。
本文提出一种神经网络算法，可从数据中自动识别相似性关系。通过揭示这些关系，网络近似描述无量纲量与其无量纲变量和系数之间的底层物理规律。同时，我们构建了一个线性代数框架，并提供代码，用于推导与这些相似性关系相关的对称群。该方法具有通用性，通过流体力学中的多个案例进行了验证，包括光滑管道中的层流牛顿流体与非牛顿流体，以及光滑和粗糙管道中的湍流。这些例子展示了框架在处理简单与复杂情形下的能力，进一步验证了其从数据中发现底层物理规律的有效性。
原文摘要 · Abstract (English)
In this work, we introduce a neural network algorithm designed to automatically identify similarity relations from data. By uncovering these similarity relations, our network approximates the underlying physical laws that relate dimensionless quantities to their dimensionless variables and coefficients. Additionally, we develop a linear algebra framework, accompanied by code, to derive the symmetry groups associated with these similarity relations. While our approach is general, we illustrate its application through examples in fluid mechanics, including laminar Newtonian and non-Newtonian flows in smooth pipes, as well as turbulent flows in both smooth and rough pipes. Such examples are chosen to highlight the framework's capability to handle both simple and intricate cases, and further validates its effectiveness in discovering underlying physical laws from data.
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