# Shape Constraints in Symbolic Regression using Penalized Least Squares

arXiv: https://arxiv.org/abs/2405.20800 · HTML: https://mlsanity.com/paper/2405.20800/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 20 references that are in the map.
- First posted on arXiv 31 May 2024.

## Builds on (most cited first, 6 of 6)

- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Shape-constrained Symbolic Regression -- Improving Extrapolation with Prior Knowledge](https://mlsanity.com/paper/2103.15624/) (2021)
- [Symbolic Regression Driven by Training Data and Prior Knowledge](https://mlsanity.com/paper/2004.11947/) (2020)
- [Introducing Thermodynamics-Informed Symbolic Regression -- A Tool for Thermodynamic Equations of State Development](https://mlsanity.com/paper/2309.02805/) (2023)
- [Using Shape Constraints for Improving Symbolic Regression Models](https://mlsanity.com/paper/2107.09458/) (2021)

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

- [rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models](https://mlsanity.com/paper/2501.17859/) (2025)
- [Neuro-Evolutionary Approach to Physics-Aware Symbolic Regression](https://mlsanity.com/paper/2504.16503/) (2025)

## Abstract (arXiv)

We study the addition of shape constraints (SC) and their consideration during the parameter identification step of symbolic regression (SR). SC serve as a means to introduce prior knowledge about the shape of the otherwise unknown model function into SR. Unlike previous works that have explored SC in SR, we propose minimizing SC violations during parameter identification using gradient-based numerical optimization. We test three algorithm variants to evaluate their performance in identifying three symbolic expressions from synthetically generated data sets. This paper examines two benchmark scenarios: one with varying noise levels and another with reduced amounts of training data. The results indicate that incorporating SC into the expression search is particularly beneficial when data is scarce. Compared to using SC only in the selection process, our approach of minimizing violations during parameter identification shows a statistically significant benefit in some of our test cases, without being significantly worse in any instance.