# Symbolic Regression by Exhaustive Search: Reducing the Search Space Using Syntactical Constraints and Efficient Semantic Structure Deduplication

arXiv: https://arxiv.org/abs/2109.13895 · HTML: https://mlsanity.com/paper/2109.13895/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 43 references that are in the map.
- First posted on arXiv 28 Sep 2021.

## Builds on (most cited first, 2 of 2)

- [Hash-Based Tree Similarity and Simplification in Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2107.10640/) (2021)
- [A multilevel block building algorithm for fast modeling generalized separable systems](https://mlsanity.com/paper/1706.02281/) (2017)

## Built on by (most cited first, 17 of 17)

- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge](https://mlsanity.com/paper/2209.06257/) (2022)
- [Exhaustive Symbolic Regression](https://mlsanity.com/paper/2211.11461/) (2022)
- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [Call for Action: towards the next generation of symbolic regression benchmark](https://mlsanity.com/paper/2505.03977/) (2025)
- [Priors for symbolic regression](https://mlsanity.com/paper/2304.06333/) (2023)
- [Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models](https://mlsanity.com/paper/2509.03036/) (2025)
- [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)
- [Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles](https://mlsanity.com/paper/2209.06454/) (2022)
- [Symbolic Regression with Fast Function Extraction and Nonlinear Least Squares Optimization](https://mlsanity.com/paper/2209.09675/) (2022)
- [A Functional Analysis Approach to Symbolic Regression](https://mlsanity.com/paper/2402.06299/) (2024)
- [Pulling the Carpet Below the Learner's Feet: Genetic Algorithm To Learn Ensemble Machine Learning Model During Concept Drift](https://mlsanity.com/paper/2412.09035/) (2024)
- [Equality Graph Assisted Symbolic Regression](https://mlsanity.com/paper/2511.01009/) (2025)
- [Scientific machine learning in Hydrology: a unified perspective](https://mlsanity.com/paper/2506.06308/) (2025)
- [A Comparative Study of Model Selection Criteria for Symbolic Regression](https://mlsanity.com/paper/2605.11233/) (2026)
- [Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System](https://mlsanity.com/paper/2607.13608/) (2026)

## Abstract (arXiv)

Symbolic regression is a powerful system identification technique in industrial scenarios where no prior knowledge on model structure is available. Such scenarios often require specific model properties such as interpretability, robustness, trustworthiness and plausibility, that are not easily achievable using standard approaches like genetic programming for symbolic regression. In this chapter we introduce a deterministic symbolic regression algorithm specifically designed to address these issues. The algorithm uses a context-free grammar to produce models that are parameterized by a non-linear least squares local optimization procedure. A finite enumeration of all possible models is guaranteed by structural restrictions as well as a caching mechanism for detecting semantically equivalent solutions. Enumeration order is established via heuristics designed to improve search efficiency. Empirical tests on a comprehensive benchmark suite show that our approach is competitive with genetic programming in many noiseless problems while maintaining desirable properties such as simple, reliable models and reproducibility.