# Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws

arXiv: https://arxiv.org/abs/2303.03192 · HTML: https://mlsanity.com/paper/2303.03192/

## Key facts

- Cited by 55 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 70 references that are in the map.
- First posted on arXiv 6 Mar 2023.

## Builds on (most cited first, 13 of 13)

- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Interpretable machine learning: definitions, methods, and applications](https://mlsanity.com/paper/1901.04592/) (2019)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Machine Learning in the Search for New Fundamental Physics](https://mlsanity.com/paper/2112.03769/) (2021)
- [Probabilistic Grammars for Equation Discovery](https://mlsanity.com/paper/2012.00428/) (2020)
- [A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge](https://mlsanity.com/paper/2209.06257/) (2022)
- [Analytical Modelling of Exoplanet Transit Specroscopy with Dimensional Analysis and Symbolic Regression](https://mlsanity.com/paper/2112.11600/) (2021)
- [SYMBA: Symbolic Computation of Squared Amplitudes in High Energy Physics with Machine Learning](https://mlsanity.com/paper/2206.08901/) (2022)
- [Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scatter](https://mlsanity.com/paper/2201.01305/) (2022)
- [On the functional form of the radial acceleration relation](https://mlsanity.com/paper/2301.04368/) (2023)

## Built on by (most cited first, 30 of 55)

- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [In Context Learning and Reasoning for Symbolic Regression with Large Language Models](https://mlsanity.com/paper/2410.17448/) (2024)
- [Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models](https://mlsanity.com/paper/2509.03036/) (2025)
- [Optimal Inflationary Potentials](https://mlsanity.com/paper/2310.16786/) (2023)
- [GFN-SR: Symbolic Regression with Generative Flow Networks](https://mlsanity.com/paper/2312.00396/) (2023)
- [Class Symbolic Regression: Gotta Fit 'Em All](https://mlsanity.com/paper/2312.01816/) (2023)
- [StruSR: Structure-Aware Symbolic Regression with Physics-Informed Taylor Guidance](https://mlsanity.com/paper/2510.06635/) (2025)
- [Boolformer: Symbolic Regression of Logic Functions with Transformers](https://mlsanity.com/paper/2309.12207/) (2023)
- [AI-Newton: A Concept-Driven Physical Law Discovery System without Prior Physical Knowledge](https://mlsanity.com/paper/2504.01538/) (2025)
- [Complexity-Aware Deep Symbolic Regression with Robust Risk-Seeking Policy Gradients](https://mlsanity.com/paper/2406.06751/) (2024)
- [Learning Interpretable Network Dynamics via Universal Neural Symbolic Regression](https://mlsanity.com/paper/2411.06833/) (2024)
- [Diffusion-Based Symbolic Regression](https://mlsanity.com/paper/2505.24776/) (2025)
- [Active Learning in Symbolic Regression with Physical Constraints](https://mlsanity.com/paper/2305.10379/) (2023)
- [Alpha Zero for Physics: Application of Symbolic Regression with Alpha Zero to find the analytical methods in physics](https://mlsanity.com/paper/2311.12713/) (2023)
- [Ab Initio Nonparametric Variable Selection for Scalable Symbolic Regression with Large p](https://mlsanity.com/paper/2410.13681/) (2024)
- [Deep Symbolic Optimization: Reinforcement Learning for Symbolic Mathematics](https://mlsanity.com/paper/2505.10762/) (2025)
- [Automated Model Discovery via Multi-modal & Multi-step Pipeline](https://mlsanity.com/paper/2509.25946/) (2025)
- [Generalizing the SINDy approach with nested neural networks](https://mlsanity.com/paper/2404.15742/) (2024)
- [Discovering an interpretable mathematical expression for a full wind-turbine wake with artificial intelligence enhanced symbolic regression](https://mlsanity.com/paper/2406.00695/) (2024)
- [An Efficient and Generalizable Symbolic Regression Method for Time Series Analysis](https://mlsanity.com/paper/2409.03986/) (2024)
- [Sparse Interpretable Deep Learning with LIES Networks for Symbolic Regression](https://mlsanity.com/paper/2506.08267/) (2025)
- [Neural non-canonical Hamiltonian dynamics for long-time simulations](https://mlsanity.com/paper/2510.01788/) (2025)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Discovering Nuclear Models from Symbolic Machine Learning](https://mlsanity.com/paper/2404.11477/) (2024)
- [A Personalised Learning Tool for Physics Undergraduate Students Built On a Large Language Model for Symbolic Regression](https://mlsanity.com/paper/2407.00065/) (2024)
- [Discovering Car-following Dynamics from Trajectory Data through Deep Learning](https://mlsanity.com/paper/2408.00251/) (2024)
- [Integrating Symbolic Neural Networks with Building Physics: A Study and Proposal](https://mlsanity.com/paper/2411.00800/) (2024)
- [Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysis](https://mlsanity.com/paper/2411.15919/) (2024)

## Abstract (arXiv)

Symbolic Regression is the study of algorithms that automate the search for analytic expressions that fit data. While recent advances in deep learning have generated renewed interest in such approaches, the development of symbolic regression methods has not been focused on physics, where we have important additional constraints due to the units associated with our data. Here we present Phi-SO, a Physical Symbolic Optimization framework for recovering analytical symbolic expressions from physics data using deep reinforcement learning techniques by learning units constraints. Our system is built, from the ground up, to propose solutions where the physical units are consistent by construction. This is useful not only in eliminating physically impossible solutions, but because the "grammatical" rules of dimensional analysis restrict enormously the freedom of the equation generator, thus vastly improving performance. The algorithm can be used to fit noiseless data, which can be useful for instance when attempting to derive an analytical property of a physical model, and it can also be used to obtain analytical approximations to noisy data. We test our machinery on a standard benchmark of equations from the Feynman Lectures on Physics and other physics textbooks, achieving state-of-the-art performance in the presence of noise (exceeding 0.1%) and show that it is robust even in the presence of substantial (10%) noise. We showcase its abilities on a panel of examples from astrophysics.