# Symbolic Regression via Control Variable Genetic Programming

arXiv: https://arxiv.org/abs/2306.08057 · HTML: https://mlsanity.com/paper/2306.08057/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 60 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Thinking Fast and Slow with Deep Learning and Tree Search](https://mlsanity.com/paper/1705.08439/) (2017)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Thinking Fast and Slow in AI](https://mlsanity.com/paper/2010.06002/) (2020)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery](https://mlsanity.com/paper/2206.10540/) (2022)
- [Learning to branch with Tree MDPs](https://mlsanity.com/paper/2205.11107/) (2022)
- [Neural Program Synthesis with Priority Queue Training](https://mlsanity.com/paper/1801.03526/) (2018)
- [Taylor Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2205.09751/) (2022)
- [Elite Bases Regression: A Real-time Algorithm for Symbolic Regression](https://mlsanity.com/paper/1704.07313/) (2017)
- [Neural-Network-Directed Genetic Programmer for Discovery of Governing Equations](https://mlsanity.com/paper/2203.08808/) (2022)

## Built on by (most cited first, 9 of 9)

- [Vertical Symbolic Regression via Deep Policy Gradient](https://mlsanity.com/paper/2402.00254/) (2024)
- [Racing Control Variable Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2309.07934/) (2023)
- [Closed-form Solutions: A New Perspective on Solving Differential Equations](https://mlsanity.com/paper/2405.14620/) (2024)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Vertical Symbolic Regression](https://mlsanity.com/paper/2312.11955/) (2023)
- [SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving](https://mlsanity.com/paper/2602.03816/) (2026)
- [Symbolic Density Estimation: A Decompositional Approach](https://mlsanity.com/paper/2603.27955/) (2026)
- [Identifying Unknown Stochastic Dynamics via Finite expression methods](https://mlsanity.com/paper/2504.07085/) (2025)
- [Prompting Neural-Guided Equation Discovery Based on Residuals](https://mlsanity.com/paper/2511.05586/) (2025)

## Abstract (arXiv)

Learning symbolic expressions directly from experiment data is a vital step in AI-driven scientific discovery. Nevertheless, state-of-the-art approaches are limited to learning simple expressions. Regressing expressions involving many independent variables still remain out of reach. Motivated by the control variable experiments widely utilized in science, we propose Control Variable Genetic Programming (CVGP) for symbolic regression over many independent variables. CVGP expedites symbolic expression discovery via customized experiment design, rather than learning from a fixed dataset collected a priori. CVGP starts by fitting simple expressions involving a small set of independent variables using genetic programming, under controlled experiments where other variables are held as constants. It then extends expressions learned in previous generations by adding new independent variables, using new control variable experiments in which these variables are allowed to vary. Theoretically, we show CVGP as an incremental building approach can yield an exponential reduction in the search space when learning a class of expressions. Experimentally, CVGP outperforms several baselines in learning symbolic expressions involving multiple independent variables.