mlsanityData to 7 Oct 2026

Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

arXiv 2305.01582 ↗First posted 2 May 2023astro-ph.IM cs.LG cs.NEcited by 36

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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.

Key facts. Cited by 36 papers in the map (data through 7 Oct 2026). Builds on 34 of its 114 references that are in the map. First posted on arXiv 2 May 2023.

Builds on

34 of 114 references.

References · 34

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2018Regularized Evolution for Image Classifier Architecture Searchcited by 902
  3. 2019AI Feynman: a Physics-Inspired Method for Symbolic Regressioncited by 536
  4. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  5. 2020Universal Differential Equations for Scientific Machine Learningcited by 345
  6. 2020Discovering Symbolic Models from Deep Learning with Inductive Biasescited by 335
  7. 2019Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradientscited by 294
  8. 2021Contemporary Symbolic Regression Methods and their Relative Performancecited by 209
  9. 2022Why do tree-based models still outperform deep learning on tabular data?cited by 198
  10. 2022End-to-end symbolic regression with transformerscited by 195
  11. 2018Learning Equations for Extrapolation and Controlcited by 151
  12. 2020AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularitycited by 140
22 more
  1. 2020AutoML-Zero: Evolving Machine Learning Algorithms From Scratchcited by 137
  2. 2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 88
  3. 2019Bayesian Symbolic Regressioncited by 77
  4. 2020AI Poincar'e: Machine Learning Conservation Laws from Trajectoriescited by 77
  5. 2019DeepMoD: Deep learning for Model Discovery in noisy datacited by 70
  6. 2022Symbolic Regression is NP-hardcited by 68
  7. 2020A Bayesian machine scientist to aid in the solution of challenging scientific problemscited by 61
  8. 2022Deep Symbolic Regression for Recurrent Sequencescited by 53
  9. 2019Learning Symbolic Physics with Graph Networkscited by 52
  10. 2018Don't Unroll Adjoint: Differentiating SSA-Form Programscited by 44
  11. 2020Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networkscited by 33
  12. 2019Epsilon-Lexicase Selection for Regressioncited by 31
  13. 2021An Approach to Symbolic Regression Using Feyncited by 20
  14. 2022Evolving symbolic density functionalscited by 19
  15. 2021Analytical Modelling of Exoplanet Transit Specroscopy with Dimensional Analysis and Symbolic Regressioncited by 18
  16. 2018A Greedy Search Tree Heuristic for Symbolic Regressioncited by 17
  17. 2019Data-driven discovery of free-form governing differential equationscited by 16
  18. 2021AI Descartes: Combining Data and Theory for Derivable Scientific Discoverycited by 7
  19. 2021High-performance symbolic-numerics via multiple dispatchcited by 6
  20. 2023Interpretable Symbolic Regression for Data Science: Analysis of the 2022 Competitioncited by 6
  21. 2022Automated discovery of interpretable gravitational-wave population modelscited by 5
  22. 2022The SZ flux-mass (Y-M) relation at low halo masses: improvements with symbolic regression and strong constraints on baryonic feedbackcited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 6; 2024: 5; 2025: 13; 2026: 12.

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  1. 2025NewtonBench: Benchmarking Generalizable Scientific Law Discovery in LLM Agentscited by 32
  2. 2023AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identificationcited by 29
  3. 2025NeuralFoil: An Airfoil Aerodynamics Analysis Tool Using Physics-Informed Machine Learningcited by 11
  4. 2023Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challengescited by 9
  5. 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
  6. 2024In Context Learning and Reasoning for Symbolic Regression with Large Language Modelscited by 7
  7. 2023Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discoverycited by 6
  8. 2026Model Discovery Agent: LLM-assisted Bayesian experiment design for data-efficient discovery of mechanistic world modelscited by 6
  9. 2025Foundation Model Self-Play: Open-Ended Strategy Innovation via Foundation Modelscited by 5
  10. 2024Distilling Reinforcement Learning Policies for Interpretable Robot Locomotion: Gradient Boosting Machines and Symbolic Regressioncited by 4
  11. 2025T-SHRED: Symbolic Regression for Regularization and Model Discovery with Transformer Shallow Recurrent Decoderscited by 4
  12. 2024COST CA20120 INTERACT Framework of Artificial Intelligence Based Channel Modelingcited by 3
24 more
  1. 2025Kozax: Flexible and Scalable Genetic Programming in JAXcited by 3
  2. 2025Symbolic Regression with Multimodal Large Language Models and Kolmogorov Arnold Networkscited by 3
  3. 2023Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculuscited by 2
  4. 2024Hyperdimensional computing: a fast, robust and interpretable paradigm for biological datacited by 2
  5. 2024Discovering an interpretable mathematical expression for a full wind-turbine wake with artificial intelligence enhanced symbolic regressioncited by 2
  6. 2025Introduction to Symbolic Regression in the Physical Sciencescited by 2
  7. 2026An Empirical Investigation of Neural ODEs and Symbolic Regression for Dynamical Systemscited by 2
  8. 2023Discovering Symbolic Laws Directly from Trajectories with Hamiltonian Graph Neural Networkscited by 1
  9. 2025Combining Large Language Models and Gradient-Free Optimization for Automatic Control Policy Synthesiscited by 1
  10. 2025Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physicscited by 1
  11. 2026DeepInflation: an AI agent for research and model discovery of inflationcited by 1
  12. 2026Architecture-Induced Recoverability Bias in Differentiable Symbolic Regressioncited by 1
  13. 2026Discovery of Interpretable Surrogates via Agentic AI: Application to Gravitational Wavescited by 1
  14. 2025Sparse Interpretable Deep Learning with LIES Networks for Symbolic Regressioncited by 0
  15. 2025Machine-Learning-Assisted Photonic Device Development: A Multiscale Approach from Theory to Characterizationcited by 0
  16. 2025SRLR: Symbolic Regression based Logic Recovery to Counter Programmable Logic Controller Attackscited by 0
  17. 2025From GNNs to Symbolic Surrogates via Kolmogorov-Arnold Networks for Delay Predictioncited by 0
  18. 2026Test-Time Scaling for Scientific Equation Discoverycited by 0
  19. 2026FluxDisco: Symbolic Regression for Stoichiometric Dynamical Systems via Monte Carlo Graph Searchcited by 0
  20. 2026Symbolic Density Estimators for Unnormalized Distributionscited by 0
  21. 2026Coefficient Calibration as Selection Pressure in Symbolic Regressioncited by 0
  22. 2026RISR: Residual-Informed Scientific Equation Discovery with Large Language Modelscited by 0
  23. 2026An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillationcited by 0
  24. 2026Machine Learning Meets High-Energy Nuclear Physics: From Pattern Recognition to Physics-Integrated Discoverycited by 0