# Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

arXiv: https://arxiv.org/abs/2305.01582 · HTML: https://mlsanity.com/paper/2305.01582/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 114 references that are in the map.
- First posted on arXiv 2 May 2023.

## Builds on (most cited first, 30 of 34)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Regularized Evolution for Image Classifier Architecture Search](https://mlsanity.com/paper/1802.01548/) (2018)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [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)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Why do tree-based models still outperform deep learning on tabular data?](https://mlsanity.com/paper/2207.08815/) (2022)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [AutoML-Zero: Evolving Machine Learning Algorithms From Scratch](https://mlsanity.com/paper/2003.03384/) (2020)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Bayesian Symbolic Regression](https://mlsanity.com/paper/1910.08892/) (2019)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Deep Symbolic Regression for Recurrent Sequences](https://mlsanity.com/paper/2201.04600/) (2022)
- [Learning Symbolic Physics with Graph Networks](https://mlsanity.com/paper/1909.05862/) (2019)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networks](https://mlsanity.com/paper/2003.04299/) (2020)
- [Epsilon-Lexicase Selection for Regression](https://mlsanity.com/paper/1905.13266/) (2019)
- [An Approach to Symbolic Regression Using Feyn](https://mlsanity.com/paper/2104.05417/) (2021)
- [Evolving symbolic density functionals](https://mlsanity.com/paper/2203.02540/) (2022)
- [Analytical Modelling of Exoplanet Transit Specroscopy with Dimensional Analysis and Symbolic Regression](https://mlsanity.com/paper/2112.11600/) (2021)
- [A Greedy Search Tree Heuristic for Symbolic Regression](https://mlsanity.com/paper/1801.01807/) (2018)
- [Data-driven discovery of free-form governing differential equations](https://mlsanity.com/paper/1910.05117/) (2019)
- [AI Descartes: Combining Data and Theory for Derivable Scientific Discovery](https://mlsanity.com/paper/2109.01634/) (2021)

## Built on by (most cited first, 30 of 54)

- [NewtonBench: Benchmarking Generalizable Scientific Law Discovery in LLM Agents](https://mlsanity.com/paper/2510.07172/) (2025)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [NeuralFoil: An Airfoil Aerodynamics Analysis Tool Using Physics-Informed Machine Learning](https://mlsanity.com/paper/2503.16323/) (2025)
- [Influence-Guided Symbolic Regression: Scientific Discovery via LLM-Driven Equation Search with Granular Feedback](https://mlsanity.com/paper/2605.29184/) (2026)
- [Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges](https://mlsanity.com/paper/2306.05014/) (2023)
- [AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression](https://mlsanity.com/paper/2312.14237/) (2023)
- [In Context Learning and Reasoning for Symbolic Regression with Large Language Models](https://mlsanity.com/paper/2410.17448/) (2024)
- [Model Discovery Agent: LLM-assisted Bayesian experiment design for data-efficient discovery of mechanistic world models](https://mlsanity.com/paper/2608.09696/) (2026)
- [Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discovery](https://mlsanity.com/paper/2308.10283/) (2023)
- [Foundation Model Self-Play: Open-Ended Strategy Innovation via Foundation Models](https://mlsanity.com/paper/2507.06466/) (2025)
- [Distilling Reinforcement Learning Policies for Interpretable Robot Locomotion: Gradient Boosting Machines and Symbolic Regression](https://mlsanity.com/paper/2403.14328/) (2024)
- [Kozax: Flexible and Scalable Genetic Programming in JAX](https://mlsanity.com/paper/2502.03047/) (2025)
- [T-SHRED: Symbolic Regression for Regularization and Model Discovery with Transformer Shallow Recurrent Decoders](https://mlsanity.com/paper/2506.15881/) (2025)
- [COST CA20120 INTERACT Framework of Artificial Intelligence Based Channel Modeling](https://mlsanity.com/paper/2411.11798/) (2024)
- [Symbolic Regression with Multimodal Large Language Models and Kolmogorov Arnold Networks](https://mlsanity.com/paper/2505.07956/) (2025)
- [Discovering Symbolic Laws Directly from Trajectories with Hamiltonian Graph Neural Networks](https://mlsanity.com/paper/2307.05299/) (2023)
- [Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculus](https://mlsanity.com/paper/2310.06609/) (2023)
- [Hyperdimensional computing: a fast, robust and interpretable paradigm for biological data](https://mlsanity.com/paper/2402.17572/) (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)
- [Sparse Interpretable Deep Learning with LIES Networks for Symbolic Regression](https://mlsanity.com/paper/2506.08267/) (2025)
- [Combining Large Language Models and Gradient-Free Optimization for Automatic Control Policy Synthesis](https://mlsanity.com/paper/2510.00373/) (2025)
- [Introduction to Symbolic Regression in the Physical Sciences](https://mlsanity.com/paper/2512.15920/) (2025)
- [An Empirical Investigation of Neural ODEs and Symbolic Regression for Dynamical Systems](https://mlsanity.com/paper/2601.20637/) (2026)
- [Discovery of Nonlinear Dynamics with Automated Basis Function Generation](https://mlsanity.com/paper/2605.09696/) (2026)
- [MaD Physics: Evaluating information seeking under constraints in physical environments](https://mlsanity.com/paper/2605.10820/) (2026)
- [Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physics](https://mlsanity.com/paper/2510.20453/) (2025)
- [DeepInflation: an AI agent for research and model discovery of inflation](https://mlsanity.com/paper/2601.14288/) (2026)
- [Symbolic Density Estimation: A Decompositional Approach](https://mlsanity.com/paper/2603.27955/) (2026)
- [Architecture-Induced Recoverability Bias in Differentiable Symbolic Regression](https://mlsanity.com/paper/2604.23256/) (2026)
- [Back to the Beginning of Heuristic Design: Bridging Code and Knowledge with LLMs](https://mlsanity.com/paper/2605.06123/) (2026)

## Abstract (arXiv)

PySR is an open-source library for practical symbolic regression, a type of machine learning which aims to discover human-interpretable symbolic models. PySR was developed to democratize and popularize symbolic regression for the sciences, and is built on a high-performance distributed back-end, a flexible search algorithm, and interfaces with several deep learning packages. PySR's internal search algorithm is a multi-population evolutionary algorithm, which consists of a unique evolve-simplify-optimize loop, designed for optimization of unknown scalar constants in newly-discovered empirical expressions. PySR's backend is the extremely optimized Julia library SymbolicRegression.jl, which can be used directly from Julia. It is capable of fusing user-defined operators into SIMD kernels at runtime, performing automatic differentiation, and distributing populations of expressions to thousands of cores across a cluster. In describing this software, we also introduce a new benchmark, "EmpiricalBench," to quantify the applicability of symbolic regression algorithms in science. This benchmark measures recovery of historical empirical equations from original and synthetic datasets.