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arxiv: 2512.15920 · v1 · submitted 2025-12-17 · 💻 cs.LG · astro-ph.IM· cs.NE· physics.comp-ph· physics.data-an

Introduction to Symbolic Regression in the Physical Sciences

Deaglan J. Bartlett , Harry Desmond , Pedro G. Ferreira , Gabriel Kronberger This is my paper

Pith reviewed 2026-05-16 21:19 UTC · model grok-4.3

classification 💻 cs.LG astro-ph.IMcs.NEphysics.comp-phphysics.data-an
keywords symbolic regressionequation discoveryphysical sciencesinterpretable modelssurrogate modelingscientific discoverygenetic programmingeffective theories
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The pith

Symbolic regression uncovers interpretable mathematical relationships directly from physical data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces symbolic regression as a method that searches for mathematical expressions fitting observed data without assuming their form in advance. It contrasts the approach with conventional regression techniques and surveys its use in the physical sciences for deriving effective theories, modeling emergent phenomena, and building compact surrogate models for expensive simulations. A sympathetic reader would care because the method can produce human-readable equations that support scientific insight while reducing reliance on opaque numerical models. The review outlines practical steps such as designing search spaces, selecting operators, controlling complexity to limit overfitting, and incorporating physical priors like symmetries. It also addresses ongoing issues including scalability to large datasets, robustness against noise, and computational cost, while pointing toward hybrids with modern AI techniques.

Core claim

Symbolic regression (SR) has emerged as a powerful method for uncovering interpretable mathematical relationships from data, offering a novel route to both scientific discovery and efficient empirical modelling. The contributions collected here span applications from automated equation discovery and emergent-phenomena modelling to the construction of compact emulators for computationally expensive simulations. The introductory review outlines the conceptual foundations of SR, contrasts it with conventional regression approaches, and surveys its main use cases in the physical sciences, including the derivation of effective theories, empirical functional forms and surrogate models. Methodicaly

What carries the argument

Symbolic regression, a search procedure that combines variables and mathematical operators into candidate expressions and selects those that best fit the data, typically via evolutionary or gradient-based algorithms.

If this is right

  • Derivation of effective theories directly from observational or simulation data in physics.
  • Construction of compact, interpretable emulators that accelerate expensive numerical simulations.
  • Data-driven modeling of emergent phenomena that lack closed-form analytic descriptions.
  • Incorporation of known physical constraints such as symmetries or asymptotic behaviors to improve search efficiency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • SR pipelines could be benchmarked on historical experimental datasets whose governing laws are already established to measure recovery accuracy.
  • Hybrid systems that first use neural networks for high-accuracy prediction and then apply SR to extract simplified equations from those predictions may yield both speed and interpretability.
  • The approach could help simplify large-scale simulation outputs in fields such as fluid dynamics or cosmology by distilling key relationships into low-complexity expressions.

Load-bearing premise

Search algorithms can reliably recover the true underlying functional form from noisy or incomplete data without excessive overfitting or computational intractability.

What would settle it

Generate synthetic datasets from a known physical equation such as Kepler's third law, add realistic measurement noise, and test whether symbolic regression recovers the original expression or consistently returns unrelated or overly complex alternatives.

read the original abstract

Symbolic regression (SR) has emerged as a powerful method for uncovering interpretable mathematical relationships from data, offering a novel route to both scientific discovery and efficient empirical modelling. This article introduces the Special Issue on Symbolic Regression for the Physical Sciences, motivated by the Royal Society discussion meeting held in April 2025. The contributions collected here span applications from automated equation discovery and emergent-phenomena modelling to the construction of compact emulators for computationally expensive simulations. The introductory review outlines the conceptual foundations of SR, contrasts it with conventional regression approaches, and surveys its main use cases in the physical sciences, including the derivation of effective theories, empirical functional forms and surrogate models. We summarise methodological considerations such as search-space design, operator selection, complexity control, feature selection, and integration with modern AI approaches. We also highlight ongoing challenges, including scalability, robustness to noise, overfitting and computational complexity. Finally we emphasise emerging directions, particularly the incorporation of symmetry constraints, asymptotic behaviour and other theoretical information. Taken together, the papers in this Special Issue illustrate the accelerating progress of SR and its growing relevance across the physical sciences.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript is an introductory review for the Special Issue on Symbolic Regression for the Physical Sciences, motivated by a Royal Society discussion meeting in April 2025. It claims that symbolic regression (SR) has emerged as a powerful method for uncovering interpretable mathematical relationships from data, offering a novel route to scientific discovery and efficient empirical modelling. The review outlines conceptual foundations of SR, contrasts it with conventional regression, surveys use cases in the physical sciences (automated equation discovery, emergent-phenomena modelling, compact emulators), summarises methodological considerations (search-space design, operator selection, complexity control, feature selection, AI integration), highlights ongoing challenges (scalability, noise robustness, overfitting, computational complexity), and emphasises emerging directions such as symmetry constraints and asymptotic behaviour.

Significance. As an introductory synthesis rather than a research contribution, the paper's value is in framing the special issue and providing a balanced entry point to SR methods for physical scientists. It accurately reflects standard conceptual foundations from prior work, explicitly flags limitations without overclaiming reliability of search algorithms, and positions the collected papers as illustrations of accelerating progress. This contextual overview supports the journal's special-issue format by helping readers navigate applications from effective theories to surrogate models.

minor comments (2)
  1. Abstract: the statement that contributions 'span applications from automated equation discovery and emergent-phenomena modelling to the construction of compact emulators' would be strengthened by indicating the approximate number of papers in the issue or listing one or two representative titles/themes to better orient readers.
  2. The review introduces terms such as 'search-space design', 'complexity control', and 'symmetry constraints' in the methodological summary without immediate concrete examples drawn from the physical sciences; inserting one brief illustrative case (e.g., operator choice in fluid-dynamics modelling) would improve accessibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript as a balanced introductory synthesis for the Special Issue. We appreciate the recommendation for minor revision. No specific major comments were raised in the report, so we have no points requiring detailed response or changes at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is an introductory review for a special issue. It surveys conceptual foundations, use cases, methodological considerations and challenges in symbolic regression without presenting any new derivations, equations, predictions or fitted quantities. No load-bearing steps reduce to self-definition, fitted inputs renamed as predictions, or self-citation chains. The central claim is presented as motivation drawn from the collected papers rather than a standalone result derived within the text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This review paper introduces no new free parameters, axioms, or invented entities; it discusses existing methods in symbolic regression without adding any novel formal structure.

pith-pipeline@v0.9.0 · 5507 in / 1029 out tokens · 20196 ms · 2026-05-16T21:19:48.710382+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Model-independent constraints on generalized FLRW consistency relations with bootstrap-based symbolic regression

    astro-ph.CO 2026-04 unverdicted novelty 6.0

    Bootstrap-based symbolic regression on supernova and BAO data finds mild 2-4 sigma deviations from FLRW consistency relations, which if real would rule out most FLRW-based solutions to cosmological tensions.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    1977 Automated theory formation in mathematics

    Lenat DB. 1977 Automated theory formation in mathematics. InProceedings of the 5th International Joint Conference on Artificial Intelligence - Volume 2IJCAI’77 pp. 833–842 San Francisco, CA, USA. Morgan Kaufmann Publishers Inc

    work page 1977

  2. [2]

    1981 BACON.5: the discovery of conservation laws

    Langley P , Bradshaw GL, Simon HA. 1981 BACON.5: the discovery of conservation laws. In Proceedings of the 7th International Joint Conference on Artificial Intelligence - Volume 1IJCAI’81 pp. 121–126 San Francisco, CA, USA. Morgan Kaufmann Publishers Inc

    work page 1981

  3. [3]

    1986Machine Learning1, 367–401

    Falkenhainer BC, Michalski RS. 1986Machine Learning1, 367–401. (10.1023/a:1022866732136)

    work page doi:10.1023/a:1022866732136

  4. [4]

    1995 Discovering dynamics: From inductive logic programming to machine discovery.Journal of Intelligent Information Systems4, 89–108

    Dzeroski S, Todorovski L. 1995 Discovering dynamics: From inductive logic programming to machine discovery.Journal of Intelligent Information Systems4, 89–108. (10.1007/bf00962824)

    work page doi:10.1007/bf00962824 1995

  5. [5]

    1992Genetic Programming: On the Programming of Computers by Means of Natural Selection

    Koza JR. 1992Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press

    work page

  6. [6]

    Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

    Cranmer M. 2023 Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl. (10.48550/ARXIV .2305.01582)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv 2023

  7. [7]

    Operon c++: An efficient genetic programming framework for symbolic regression,

    Burlacu B, Kronberger G, Kommenda M. 2020 Operon C++: an efficient genetic programming framework for symbolic regression. InProceedings of the 2020 Genetic and Evolutionary Computation Conference CompanionGECCO ’20 p. 1562–1570 New York, NY, USA. Association for Computing Machinery. (10.1145/3377929.3398099)

    work page doi:10.1145/3377929.3398099 2020

  8. [8]

    Udrescu and M

    Udrescu SM, Tegmark M. 2020 AI Feynman: A physics-inspired method for symbolic regression.Science Advances6, eaay2631. (10.1126/sciadv.aay2631) 8royalsocietypublishing.org/journal/rspa Proc R Soc A 0000000

    work page doi:10.1126/sciadv.aay2631 2020

  9. [9]

    2023 Combining data and theory for derivable scientific discovery with AI-Descartes.Nature Communications14

    Cornelio C, Dash S, Austel V , Josephson TR, Goncalves J, Clarkson KL, Megiddo N, El Khadir B, Horesh L. 2023 Combining data and theory for derivable scientific discovery with AI-Descartes.Nature Communications14. (10.1038/s41467-023-37236-y)

    work page doi:10.1038/s41467-023-37236-y 2023

  10. [10]

    2018 Learning Equations for Extrapolation and Control

    Sahoo S, Lampert C, Martius G. 2018 Learning Equations for Extrapolation and Control. In Dy J, Krause A, editors,Proceedings of the 35th International Conference on Machine Learning vol. 80Proceedings of Machine Learning Researchpp. 4442–4450. PMLR

    work page 2018

  11. [11]

    2022 A Unified Framework for Deep Symbolic Regression

    Landajuela M, Lee CS, Yang J, Glatt R, Santiago CP , Aravena I, Mundhenk T, Mulcahy G, Petersen BK. 2022 A Unified Framework for Deep Symbolic Regression. In Koyejo S, Mohamed S, Agarwal A, Belgrave D, Cho K, Oh A, editors,Advances in Neural Information Processing Systemsvol. 35 pp. 33985–33998. Curran Associates, Inc

    work page 2022