# Probabilistic Grammars for Equation Discovery

arXiv: https://arxiv.org/abs/2012.00428 · HTML: https://mlsanity.com/paper/2012.00428/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2020.

## Built on by (most cited first, 30 of 36)

- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws](https://mlsanity.com/paper/2303.03192/) (2023)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)
- [Automated Scientific Discovery: From Equation Discovery to Autonomous Discovery Systems](https://mlsanity.com/paper/2305.02251/) (2023)
- [LLM4ED: Large Language Models for Automatic Equation Discovery](https://mlsanity.com/paper/2405.07761/) (2024)
- [Efficient Generator of Mathematical Expressions for Symbolic Regression](https://mlsanity.com/paper/2302.09893/) (2023)
- [Priors for symbolic regression](https://mlsanity.com/paper/2304.06333/) (2023)
- [Accelerating Understanding of Scientific Experiments with End to End Symbolic Regression](https://mlsanity.com/paper/2112.04023/) (2021)
- [Class Symbolic Regression: Gotta Fit 'Em All](https://mlsanity.com/paper/2312.01816/) (2023)
- [Vertical Symbolic Regression via Deep Policy Gradient](https://mlsanity.com/paper/2402.00254/) (2024)
- [Grammar-based Ordinary Differential Equation Discovery](https://mlsanity.com/paper/2504.02630/) (2025)
- [Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching](https://mlsanity.com/paper/2409.01416/) (2024)
- [Automated Model Discovery via Multi-modal & Multi-step Pipeline](https://mlsanity.com/paper/2509.25946/) (2025)
- [A Reinforcement Learning Approach to Domain-Knowledge Inclusion Using Grammar Guided Symbolic Regression](https://mlsanity.com/paper/2202.04367/) (2022)
- [Generalizing the SINDy approach with nested neural networks](https://mlsanity.com/paper/2404.15742/) (2024)
- [Approximating the universal thermal climate index using sparse regression with orthogonal polynomials](https://mlsanity.com/paper/2508.11307/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Predicting symbolic ODEs from multiple trajectories](https://mlsanity.com/paper/2510.23295/) (2025)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Neuro-Symbolic ODE Discovery with Latent Grammar Flow](https://mlsanity.com/paper/2604.16232/) (2026)
- [Probabilistic Regular Tree Priors for Scientific Symbolic Reasoning](https://mlsanity.com/paper/2306.08506/) (2023)
- [Vertical Symbolic Regression](https://mlsanity.com/paper/2312.11955/) (2023)
- [Symbolic Density Estimation for Discrete Distributions](https://mlsanity.com/paper/2605.21813/) (2026)
- [VESTA: Visual Exploration with Statistical Tool Agents](https://mlsanity.com/paper/2606.00384/) (2026)
- [P(Expression|Grammar): Probability of deriving an algebraic expression with a probabilistic context-free grammar](https://mlsanity.com/paper/2212.00751/) (2022)
- [Quantifying Behavioral Dissimilarity Between Mathematical Expressions](https://mlsanity.com/paper/2408.11515/) (2024)
- [Neural-Guided Equation Discovery](https://mlsanity.com/paper/2503.16953/) (2025)
- [A Comparative Study of Model Selection Criteria for Symbolic Regression](https://mlsanity.com/paper/2605.11233/) (2026)

## Abstract (arXiv)

Equation discovery, also known as symbolic regression, is a type of automated modeling that discovers scientific laws, expressed in the form of equations, from observed data and expert knowledge. Deterministic grammars, such as context-free grammars, have been used to limit the search spaces in equation discovery by providing hard constraints that specify which equations to consider and which not. In this paper, we propose the use of probabilistic context-free grammars in equation discovery. Such grammars encode soft constraints, specifying a prior probability distribution on the space of possible equations. We show that probabilistic grammars can be used to elegantly and flexibly formulate the parsimony principle, that favors simpler equations, through probabilities attached to the rules in the grammars. We demonstrate that the use of probabilistic, rather than deterministic grammars, in the context of a Monte-Carlo algorithm for grammar-based equation discovery, leads to more efficient equation discovery. Finally, by specifying prior probability distributions over equation spaces, the foundations are laid for Bayesian approaches to equation discovery.