物理学方程中的统计模式与自然法则的涌现
Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature
AI总结：
研究发现物理方程中数学运算符频率呈指数衰减，揭示了自然法则的统计规律，为符号回归和语言模型生成数学表达式提供新思路。
摘要：
物理学试图揭示自然法则并用数学方程来表达。尽管自然现象的多样性巨大，物理方程却表现出独特的结构性规律，使其区别于任意的数学表达式。虽然尺寸分析等原则长期以来指导了物理模型的制定，但探索物理方程中更微妙的统计模式仍是一个开放性问题。在此，通过分析四个物理方程语料库并应用先进的隐式似然技术，我们发现数学运算符的频率遵循指数衰减定律，与自然语言中词频的Zipf幂律不同。这揭示了物理的统计元法则，可能反映了通信效率和自然本身施加的约束的结合。该元法则为符号回归提供了实际好处，大幅缩小了物理合理表达的空间。更广泛地说，它可能指导能够生成连贯数学表示的语言模型的发展，推动物理法则发现的自动化。
英文摘要：
Physics seeks to uncover the laws of Nature and express them through mathematical equations. Despite the vast diversity of natural phenomena, physical equations exhibit structural regularities that set them apart from arbitrary mathematical expressions. While principles such as dimensional analysis have long guided the formulation of physical models, the exploration of more subtle statistical patterns within the equations of physics remains an open question. Here, by analysing four corpora of physics equations and applying advanced implicit-likelihood techniques, we find that the frequency of mathematical operators follows an exponential decay law, in contrast to Zipf's power law for word frequencies in natural languages. This reveals a statistical meta-law of physics, possibly reflecting a combination of communication efficiency and constraints imposed by Nature itself. The meta-law offers practical benefits for symbolic regression by drastically narrowing down the space of physically plausible expressions. More broadly, it may inform the development of language models that can generate coherent mathematical representations, advancing the automation of physical law discovery.
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