模仿科学发现过程,分步推导物理定律,提升模型泛化能力
Data-driven Progressive Discovery of Physical Laws
- 将物理定律发现建模为逐步组合有意义知识单元的链条
- 在三种流体与激光-金属问题中改进经典尺度理论
- 适合需要可解释性物理模型的工程与科研场景
符号回归是知识发现的强大工具,可直接从数据中提取可解释的数学表达式。但传统方法采用一次性、端到端的处理方式,在真实物理系统中常产生冗长且无物理意义的表达式,导致模型泛化能力差。这本质上源于其偏离了科学发现的基本路径:物理定律并非单一形式存在,而是遵循从简单到复杂的层级递进规律。受此启发,我们提出链式符号回归(CoSR)框架,将物理定律发现建模为一系列具有明确物理意义的知识单元按特定逻辑逐步组合的知识链,从而实现从数据中精准发现底层物理规律。CoSR完整复现了从开普勒第三定律到万有引力定律的经典发现路径,并应用于湍流瑞利-贝纳德对流、圆管内粘性流动以及激光-金属相互作用三类问题,显著改进了经典尺度理论。最后,该方法在不同飞机气动系数尺度规律这一复杂工程问题中展现出发现新知识的能力。
原文摘要 · Abstract (English)
Symbolic regression is a powerful tool for knowledge discovery, enabling the extraction of interpretable mathematical expressions directly from data. However, conventional symbolic discovery typically follows an end-to-end, "one-step" process, which often generates lengthy and physically meaningless expressions when dealing with real physical systems, leading to poor model generalization. This limitation fundamentally stems from its deviation from the basic path of scientific discovery: physical laws do not exist in a single form but follow a hierarchical and progressive pattern from simplicity to complexity. Motivated by this principle, we propose Chain of Symbolic Regression (CoSR), a novel framework that models the discovery of physical laws as a chain of symbolic knowledge. This knowledge chain is formed by progressively combining multiple knowledge units with clear physical meanings along a specific logic, ultimately enabling the precise discovery of the underlying physical laws from data. CoSR fully recapitulates the progressive discovery path from Kepler's third law to the law of universal gravitation in classical mechanics, and is applied to three types of problems: turbulent Rayleigh-Benard convection, viscous flows in a circular pipe, and laser-metal interaction, demonstrating its ability to improve classical scaling theories. Finally, CoSR showcases its capability to discover new knowledge in the complex engineering problem of aerodynamic coefficients scaling for different aircraft.
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