# (Exhaustive) Symbolic Regression and model selection by minimum description length

arXiv: https://arxiv.org/abs/2507.13033 · HTML: https://mlsanity.com/paper/2507.13033/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 34 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 9 of 9)

- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Minimum Description Length Revisited](https://mlsanity.com/paper/1908.08484/) (2019)
- [Exhaustive Symbolic Regression](https://mlsanity.com/paper/2211.11461/) (2022)
- [An Approach to Symbolic Regression Using Feyn](https://mlsanity.com/paper/2104.05417/) (2021)
- [Priors for symbolic regression](https://mlsanity.com/paper/2304.06333/) (2023)
- [Optimal Inflationary Potentials](https://mlsanity.com/paper/2310.16786/) (2023)
- [syren-new: Precise formulae for the linear and nonlinear matter power spectra with massive neutrinos and dynamical dark energy](https://mlsanity.com/paper/2410.14623/) (2024)
- [On the functional form of the radial acceleration relation](https://mlsanity.com/paper/2301.04368/) (2023)

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

- [Dynamic Bayesian Optimization Framework for Instruction Tuning in Partial Differential Equation Discovery](https://mlsanity.com/paper/2601.00088/) (2025)
- [On the Generalization Bounds of Symbolic Regression with Genetic Programming](https://mlsanity.com/paper/2604.17402/) (2026)

## Abstract (arXiv)

Symbolic regression is the machine learning method for learning functions from data. After a brief overview of the symbolic regression landscape, I will describe the two main challenges that traditional algorithms face: they have an unknown (and likely significant) probability of failing to find any given good function, and they suffer from ambiguity and poorly-justified assumptions in their function-selection procedure. To address these I propose an exhaustive search and model selection by the minimum description length principle, which allows accuracy and complexity to be directly traded off by measuring each in units of information. I showcase the resulting publicly available Exhaustive Symbolic Regression algorithm on three open problems in astrophysics: the expansion history of the universe, the effective behaviour of gravity in galaxies and the potential of the inflaton field. In each case the algorithm identifies many functions superior to the literature standards. This general purpose methodology should find widespread utility in science and beyond.