# Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature

arXiv: https://arxiv.org/abs/2408.11065 · HTML: https://mlsanity.com/paper/2408.11065/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 59 references that are in the map.
- First posted on arXiv 12 Aug 2024.

## Builds on (most cited first, 13 of 13)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Sampling-Based Accuracy Testing of Posterior Estimators for General Inference](https://mlsanity.com/paper/2302.03026/) (2023)
- [Exhaustive Symbolic Regression](https://mlsanity.com/paper/2211.11461/) (2022)
- [Studying the Difference Between Natural and Programming Language Corpora](https://mlsanity.com/paper/1806.02437/) (2018)
- [Evidence Networks: simple losses for fast, amortized, neural Bayesian model comparison](https://mlsanity.com/paper/2305.11241/) (2023)
- [LtU-ILI: An All-in-One Framework for Implicit Inference in Astrophysics and Cosmology](https://mlsanity.com/paper/2402.05137/) (2024)
- [Optimal Inflationary Potentials](https://mlsanity.com/paper/2310.16786/) (2023)
- [Zipf's law and criticality in multivariate data without fine-tuning](https://mlsanity.com/paper/1310.0448/) (2013)
- [Understanding Zipf's law of word frequencies through sample-space collapse in sentence formation](https://mlsanity.com/paper/1407.4610/) (2014)

## Built on by (most cited first, 2 of 2)

- [Explainable fault and severity classification for rolling element bearings using Kolmogorov-Arnold networks](https://mlsanity.com/paper/2412.01322/) (2024)
- [Bayesian Symbolic Regression with Entropic Reinforcement Learning](https://mlsanity.com/paper/2608.09617/) (2026)

## Abstract (arXiv)

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.