# Exhaustive Symbolic Regression

arXiv: https://arxiv.org/abs/2211.11461 · HTML: https://mlsanity.com/paper/2211.11461/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 83 references that are in the map.
- First posted on arXiv 21 Nov 2022.

## Builds on (most cited first, 16 of 16)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [PMLB: A Large Benchmark Suite for Machine Learning Evaluation and Comparison](https://mlsanity.com/paper/1703.00512/) (2017)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [PMLB v1.0: An open source dataset collection for benchmarking machine learning methods](https://mlsanity.com/paper/2012.00058/) (2020)
- [Minimum Description Length Revisited](https://mlsanity.com/paper/1908.08484/) (2019)
- [An Approach to Symbolic Regression Using Feyn](https://mlsanity.com/paper/2104.05417/) (2021)
- [Symbolic Regression by Exhaustive Search: Reducing the Search Space Using Syntactical Constraints and Efficient Semantic Structure Deduplication](https://mlsanity.com/paper/2109.13895/) (2021)
- [Complexity Measures for Multi-objective Symbolic Regression](https://mlsanity.com/paper/2109.00238/) (2021)
- [Hash-Based Tree Similarity and Simplification in Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2107.10640/) (2021)
- [The SZ flux-mass (Y-M) relation at low halo masses: improvements with symbolic regression and strong constraints on baryonic feedback](https://mlsanity.com/paper/2209.02075/) (2022)
- [A New Deterministic Technique for Symbolic Regression](https://mlsanity.com/paper/1908.06754/) (2019)
- [Learning Transport Processes with Machine Intelligence](https://mlsanity.com/paper/2109.13096/) (2021)

## Built on by (most cited first, 29 of 29)

- [Priors for symbolic regression](https://mlsanity.com/paper/2304.06333/) (2023)
- [ParFam -- (Neural Guided) Symbolic Regression Based on Continuous Global Optimization](https://mlsanity.com/paper/2310.05537/) (2023)
- [Symbolic Regression for Beyond the Standard Model Physics](https://mlsanity.com/paper/2405.18471/) (2024)
- [A Comparison of Recent Algorithms for Symbolic Regression to Genetic Programming](https://mlsanity.com/paper/2406.03585/) (2024)
- [Scaling Up Unbiased Search-based Symbolic Regression](https://mlsanity.com/paper/2506.19626/) (2025)
- [No One-Size-Fits-All Neurons: Task-based Neurons for Artificial Neural Networks](https://mlsanity.com/paper/2405.02369/) (2024)
- [COmoving Computer Acceleration (COCA): N-body simulations in an emulated frame of reference](https://mlsanity.com/paper/2409.02154/) (2024)
- [Symbolic Regression with Multimodal Large Language Models and Kolmogorov Arnold Networks](https://mlsanity.com/paper/2505.07956/) (2025)
- [Generalizing the SINDy approach with nested neural networks](https://mlsanity.com/paper/2404.15742/) (2024)
- [Statistical Patterns in the Equations of Physics and the Emergence of a Meta-Law of Nature](https://mlsanity.com/paper/2408.11065/) (2024)
- [(Exhaustive) Symbolic Regression and model selection by minimum description length](https://mlsanity.com/paper/2507.13033/) (2025)
- [Mitigating Data Scarcity in Time Series Analysis: A Foundation Model with Series-Symbol Data Generation](https://mlsanity.com/paper/2502.15466/) (2025)
- [Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physics](https://mlsanity.com/paper/2510.20453/) (2025)
- [Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Families](https://mlsanity.com/paper/2602.11630/) (2026)
- [Symbolic Density Estimation for Discrete Distributions](https://mlsanity.com/paper/2605.21813/) (2026)
- [NeuronSeek: On Stability and Expressivity of Task-driven Neurons](https://mlsanity.com/paper/2506.15715/) (2025)
- [Auditable Unit-Aware Thresholds in Symbolic Regression via Logistic-Gated Operators](https://mlsanity.com/paper/2510.05178/) (2025)
- [Symbolic Emulators for Cosmology: Accelerating Cosmological Analyses Without Sacrificing Precision](https://mlsanity.com/paper/2510.18749/) (2025)
- [Constraining dark matter halo profiles with symbolic regression](https://mlsanity.com/paper/2511.23073/) (2025)
- [Symbolic Regression for Shared Expressions: Introducing Partial Parameter Sharing](https://mlsanity.com/paper/2601.04051/) (2026)
- [On the Generalization Bounds of Symbolic Regression with Genetic Programming](https://mlsanity.com/paper/2604.17402/) (2026)
- [Symbolic Classification-Enabled LHC Limits for BSM Global Fits](https://mlsanity.com/paper/2605.22330/) (2026)
- [Are you sure? A Comprehensive and Comprehensible Survey of Uncertainty Quantification in Symbolic Regression](https://mlsanity.com/paper/2606.06567/) (2026)
- [ERBench: A Benchmark and Testsuite for Equation Discovery Algorithms](https://mlsanity.com/paper/2606.09276/) (2026)
- [The Degeneracy Distillery](https://mlsanity.com/paper/2606.23838/) (2026)
- [Evolutional Math: Cross-Validated Island-Model Genetic Programming for Interpretable Symbolic Regression on Small, Wide Datasets](https://mlsanity.com/paper/2606.28381/) (2026)
- [Evaluation of Population Initialization Methods for Genetic Programming-based Symbolic Regression](https://mlsanity.com/paper/2606.31990/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [TNLearn: An Open Source Python Package for Task-based Neurons](https://mlsanity.com/paper/2609.27564/) (2026)

## Abstract (arXiv)

Symbolic Regression (SR) algorithms attempt to learn analytic expressions which fit data accurately and in a highly interpretable manner. Conventional SR suffers from two fundamental issues which we address here. First, these methods search the space stochastically (typically using genetic programming) and hence do not necessarily find the best function. Second, the criteria used to select the equation optimally balancing accuracy with simplicity have been variable and subjective. To address these issues we introduce Exhaustive Symbolic Regression (ESR), which systematically and efficiently considers all possible equations -- made with a given basis set of operators and up to a specified maximum complexity -- and is therefore guaranteed to find the true optimum (if parameters are perfectly optimised) and a complete function ranking subject to these constraints. We implement the minimum description length principle as a rigorous method for combining these preferences into a single objective. To illustrate the power of ESR we apply it to a catalogue of cosmic chronometers and the Pantheon+ sample of supernovae to learn the Hubble rate as a function of redshift, finding ~40 functions (out of 5.2 million trial functions) that fit the data more economically than the Friedmann equation. These low-redshift data therefore do not uniquely prefer the expansion history of the standard model of cosmology. We make our code and full equation sets publicly available.