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Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

Paper · 2023 · Miles Cranmer (Princeton University and Flatiron Institute)

Summary

Symbolic regression (SR) searches over analytic expressions to find formulas that are both accurate and readable. It automates the kind of empirical law-finding that produced Kepler's third law or Planck's law. The author argues that most existing SR tools are tuned to synthetic benchmarks and fall short of what scientists need. Those needs include fitting unknown real constants, handling noisy and heteroscedastic data, piecewise forms, custom operators, constraints, high-dimensional or non-tabular inputs, and open, easy-to-use software.

To address this, the paper presents PySR, a Python package running on the Julia engine SymbolicRegression.jl. Its search is a multi-population evolutionary algorithm built around a cycle of evolving, simplifying and constant-optimizing candidate expressions. It also uses age-regularized replacement and simulated annealing. The backend compiles user-defined operators into fast SIMD code, supports automatic differentiation, and scales across cluster cores.

The paper also introduces EmpiricalBench, a benchmark that tests whether algorithms can rediscover historical empirical equations. On it, classic untrained methods outperformed pure deep-learning approaches. This suggests careful search heuristics still matter for real scientific data.

Key points

  • PySR is distributed via PyPI and Conda under pysr. Its backend, SymbolicRegression.jl, is in the Julia registry and can be used directly from Julia. It was first released publicly in 2020.
  • The algorithm modifies classic tournament-selection evolution in several ways:
    • it removes the oldest individual rather than the weakest (age-regularized evolution);
    • it adds simulated annealing, with rejection probability controlled by temperature T and hyperparameter α;
    • it runs asynchronous migration between populations.
  • EmpiricalBench measures recovery of known empirical laws from original and synthetic datasets. On it, PySR, Operon, DSR and QLattice all beat the deep-learning methods EQL and SR-Transformer. SR-Transformer is pre-trained on billions of synthetic expressions, yet still came out behind.
  • PySR has been used to model cloud cover, electron transfer in materials, astrophysical relations, international trade rules, and gravitational-wave source populations.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🎓Princeton UniversityPrinceton · United States137 works · 53,849 citations🔬Flatiron InstituteNew York · United States2 works · 235 citations

 Fields, topics and concepts

📚 Shelves 5The field of science this work belongs to (OpenAlex fields).Computer Science · 19,503 worksArtificial Intelligence · 18,805 worksPhysics and Astronomy · 70 worksStatistical and Nonlinear Physics · 60 worksNuclear and High Energy Physics · 2 works

Cite

Cite as Miles Cranmer (Princeton University and Flatiron Institute). 2023. Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl. arXiv:2305.01582.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.48550/arxiv.2305.01582 · arXiv 2305.01582

@misc{university2023interpretable,
  title = {{Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl}},
  author = {Miles Cranmer (Princeton University and Flatiron Institute)},
  year = {2023},
  doi = {10.48550/arxiv.2305.01582},
  eprint = {2305.01582},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2305.01582}
}

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