Pith. sign in

REVIEW 3 major objections 6 minor 2 cited by

Across five real-world datasets, the four public multi-view symbolic regression implementations frequently produce compact, accurate models even without fine-tuning.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Benchmarking four multi-view symbolic regression packages on five real scientific datasets shows all find accurate compact models; parameter limits and shared constants emerge as key design features.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Useful first real-world benchmark of four MvSR implementations; the central qualitative claim holds, but the comparative rankings rest on a metric with inconsistent parameter counting and should be treated as suggestive, not definitive. the 3 major comments →

arxiv 2509.10500 v1 pith:ZXDTYNKF submitted 2025-09-01 cs.LG astro-ph.GAastro-ph.IMphysics.data-an

Exploring Multi-view Symbolic Regression methods in physical sciences

Etienne Russeil , Fabr\'icio Olivetti de Fran\c{c}a , Konstantin Malanchev , Guillaume Moinard , Maxime Cherrey This is my paper
classification cs.LG astro-ph.GAastro-ph.IMphysics.data-an
keywords multi-view symbolic regressionsymbolic regressioninterpretable modelsphenomenological equationsbenchmarkparameter sparsityphysical sciences
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Multi-view symbolic regression (MvSR) searches for one parametric function that, after view-specific parameter fitting, describes several datasets produced by the same underlying phenomenon. The paper compares the four public implementations that currently support MvSR—Operon, PySR, ϕ-SO, and eggp—on five real-world physical datasets covering supernova light curves, galaxy rotation curves, Michaelis–Menten kinetics, pipe friction, and network degree distributions. Its central finding is that all four methods often produce accurate models with only a few free parameters, even under fixed hyperparameters with no expert fine-tuning. The paper does not claim a definitive winner; instead it identifies design features that matter: hard limits on the number of parameters, regulated use of constants shared across views, and an explicit choice of how per-view losses are aggregated. If correct, these results are a practical green light for using MvSR on real one-dimensional datasets and a concrete checklist for future tool development.

Core claim

The paper claims that, despite the fundamental difficulty of comparing implementations that use different algorithms, Multi-view Symbolic Regression as it exists today already delivers on its promise: across five real-world benchmarks, the MvSR variants in Operon, PySR, ϕ-SO, and eggp frequently found solutions whose training reduced mean-squared error fell below the paper's 5×10−3 'good model' threshold, and many of those solutions use two to four view-specific parameters and few global constants. The authors also claim that specific features drive success: restricting the maximum number of parameters is essential for simple, relevant models (unrestricted runs fit best on raw accuracy but a

What carries the argument

The Multi-view Symbolic Regression objective: minimize over the shared functional form f of an aggregate (max or mean) of per-view losses, where each view i minimizes its own parameters θ_i, i.e. min_f agg_i (min_{θ_i} L(f(x_i; θ_i), y_i)). This formulation is what lets a single equation absorb view-specific variation into a few reused parameters and keeps the model from overfitting any one dataset. The paper's evaluation engine is the reduced mean-squared error, MSE* = Σ(y−ŷ)^2/(n_obs − n_params), aggregated by the worst (maximum) per-view value; it is the metric that grounds the 'good model' threshold and the comparative statements.

Load-bearing premise

The comparative conclusions rest on the reduced mean-squared-error metric aggregated by worst per-view fit, which ignores measurement uncertainties, counts reused and shared constants inconsistently, and sets n_params to zero on test data; if that metric mis-orders models, the claims about which implementations generate better models lose support.

What would settle it

Recompute the benchmark with a loss that weights each point by its reported uncertainty where available, counts every distinct parameter and shared constant when computing degrees of freedom on test data, and treats Nikuradse's pipe roughness as a continuous covariate rather than as view labels. If the ranking of implementations changes materially—say, eggp no longer yields the most satisfactory models or PySR no longer the fewest—the paper's central comparison is an artifact of its metric.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Practitioners can run current MvSR tools on one-dimensional multi-dataset problems with stock hyperparameters and reasonably expect compact, accurate models that interpolate and sometimes extrapolate well.
  • MvSR developers should implement or default to a hard parameter-count limit (or a degrees-of-freedom penalty such as AIC/BIC/MDL), because unconstrained models win on accuracy but lose interpretability and physical plausibility.
  • Shared constants across views are a valuable feature, but tools need a user-set maximum; without one, constants multiply and make models hard to read, as when PySR's light-curve model needs seven constants where ϕ-SO uses one.
  • The aggregation function should be exposed as an explicit hyperparameter, with worst-fitness recommended when all views are equally important and mean when outliers or large samples are present.
  • Uncertainty handling should become a standard MvSR capability; currently only PySR and ϕ-SO accept error bars, so measurement noise is silently ignored in cross-method comparisons.

Where Pith is reading between the lines

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

  • Extension: treating Nikuradse's pipe roughness as a continuous covariate rather than as discrete view labels would remove an artificial partition the paper itself notes, and could change the ranking of methods on that dataset.
  • Extension: extending the same benchmark to multidimensional real-world inputs would test whether MvSR's parameter-sharing benefit grows with view count and input dimension, a natural next step the paper leaves open.
  • Extension: a corrected complexity metric that counts reused parameters and shared constants in n_params, instead of setting n_params to zero on test data, might reorder methods when judged on out-of-sample parsimony.
  • Extension: because only the single best returned model is judged per run, exporting and exploring the top-N solutions visited during the search, as the paper recommends, could change which tool looks most useful in practice.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper presents an empirical comparison of four symbolic regression (SR) implementations that support multi-view symbolic regression (MvSR): Operon, PySR, ϕ-SO, and eggp. The authors apply these methods to five real-world datasets from physics and network science, partitioning each dataset into multiple views and training a parametric model with view-specific parameters. They evaluate models using a reduced mean squared error (MSE*) aggregated with the worst-fitness rule, and report both training and test scores. The central claim is that all MvSR implementations 'often achieve good accuracy while proposing solutions with only few free parameters' and that certain implementation features (parameter limits, shared-constant limits, aggregation control) are desirable for future development. The paper concludes with guidelines for MvSR design and makes all code and results publicly available.

Significance. If the results hold, this is a useful and timely empirical study: MvSR is a recent extension of SR, and a comparison of the four publicly available implementations on real scientific data has not been performed before. The paper benefits from using real, domain-relevant datasets rather than synthetic benchmarks, and from releasing code and results. The authors are also transparent about several limitations, including the absence of data uncertainties, the restriction to one-dimensional problems, and the impossibility of a fully fair cross-implementation comparison. However, the quantitative conclusions—especially the comparative statements about which methods produce more satisfactory models—depend on a metric (MSE*) whose parameter-counting rules are ambiguous and applied inconsistently between training and test phases. The central qualitative message is defensible, but the load-bearing metric needs clarification or revision before the comparative claims can be accepted as stated.

major comments (3)
  1. [Section 4, Eq. (4.1)] The reduced MSE is defined as MSE/(n_obs − n_params), but the paper never specifies how n_params is counted for expressions where the same parameter symbol appears multiple times, nor for constants shared across views. The four implementations differ in exactly these respects: Operon forbids parameter reuse, PySR and eggp allow it, and ϕ-SO distinguishes view-specific parameters from globally shared constants. If n_params counts occurrences rather than distinct symbols, the complexity penalty is not comparable across methods. Compounding this, the test loss is computed with n_params = 0, so test MSE* is just raw MSE, putting training and test scores on different scales. This asymmetry can misorder models: a complex model with many reused parameters may receive a favorable training penalty and an unpenalized test score, while a genuinely simpler model is ranked lower. This directly affect
  2. [Section 5(a) and Table 2] There is an inconsistency in how the 'best' model is selected. Section 5(a) states that for each dataset and implementation the authors study 'the model with the lowest MSE_test that was produced,' yet the caption of Table 2 says the solutions are 'from the lowest MSE_train.' Figure 5's caption also says 'lowest MSE_train solution,' while other figure captions refer to 'lowest reduced MSE solution.' This inconsistency matters because the showcased models and the discussion of their qualities (e.g., PySR's galaxy model, ϕ-SO's Michaelis–Menten model) depend on which criterion was used. If the selection was based on training MSE, the 'test accuracy' claims for those models are not properly supported. Please clarify the selection criterion and ensure it is applied consistently.
  3. [Figure 1 and Section 5(a)] The global cumulative distributions pool all datasets, hyperparameter configurations, and seeds into a single histogram. Because the datasets have very different numbers of views, points per view, and intrinsic difficulty, the pooled distribution may be dominated by the easiest datasets or by the worst-fitness aggregation behavior. The statement that 'PySR finds significantly less good models' rests on this pooled comparison. A per-dataset breakdown (e.g., separate cumulative distributions or median MSE* per dataset) is needed to support the comparative claims. Without it, the observed differences could be driven by one or two datasets rather than by general method behavior.
minor comments (6)
  1. [Section 3, Nikuradse dataset] Typo: 'The data offers a with very high sampling' should likely be 'The data offers a very high sampling.'
  2. [Section 6] Typo: 'This is evidence by' should be 'This is evidenced by.'
  3. [Section 4] In the sentence 'In this case nparams = 0and M SE∗ = M SE,' there is a missing space and the notation could be formatted consistently.
  4. [Section 2, Eq. (2.1)] The definition of n as 'the number of parameters in the model (θ)' with bounds l < n < u is slightly confusing because θ is indexed per view (θ_i). Clarify that n is the number of view-specific parameters.
  5. [Figure 1] The caption states 'Medians not shown are exceeding the clipping threshold.' Consider indicating the actual median values or marking the clipping threshold more explicitly, as the reader cannot tell which medians are absent.
  6. [Table 2] Some expressions are difficult to parse due to formatting (e.g., square-root continuations and stacked fractions). Consider using a cleaner typesetting or adding a brief explanation of the notation used for constants and parameters.

Circularity Check

0 steps flagged

No material circularity: the paper is an empirical benchmark; its conclusions are direct measurements of method outputs, and the only self-citations are descriptive, not load-bearing.

full rationale

The paper does not claim to derive a prediction from a first-principles result. It runs four existing MvSR implementations on real datasets under fixed hyperparameters and reports measured reduced MSE (Eq. 4.1), node counts, runtimes, and the best expressions (Table 2, Figs. 1-7). The central statements—that methods 'often achieve good accuracy while proposing solutions with only few free parameters' and that features like parameter control or constant limits are useful—are inductive generalizations from these measurements, not consequences of the metric by construction. The self-citations, e.g. '[13]' for the MvSR loss definition (Eq. 2.1) and for the Operon extension, and reuse of the supernovae dataset from [13], describe tools and prior data; they do not supply the evidence for the benchmark findings, which are independently reproducible from the released code. The acknowledged limitations—the stated impossibility of rigorous comparison, the artificial view partition for Nikuradse, absent uncertainties, and the one-dimensional restriction—are validity caveats, not circularity. The reduced-MSE metric's parameter-counting details and the nparams=0 test-loss convention could bias rankings, but that is a measurement-validity concern; it does not make any claimed result equivalent to its input by definition. No circular step can be exhibited, so per the hard rules no circularity is flagged.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The central claims are empirical and rest on the selected datasets, the reduced-MSE metric, and the hyperparameter protocol. The most important non-standard elements are the hand-chosen MSE threshold and the assumption that worst-fitness aggregation across views is a sound strategy. No new theoretical entities are introduced.

free parameters (3)
  • Good-model threshold on reduced MSE = 5e-3
    Hand-chosen heuristic in Section 5(a) to classify models as high quality; not derived from data or theory.
  • Hyperparameter grid values = max nodes 15/30, max params 2/4, generations 100, pop 100, iterations 3, 30 min budget
    Predetermined experimental choices, not fitted to the target result, but they bound the space of discovered models and affect the central accuracy claims.
  • operon_increased setup = 1000 generations, pop 1000, 100 iterations
    Additional post hoc configuration chosen after observing Operon's poor default performance; affects conclusions about Operon's capability.
axioms (4)
  • domain assumption All views within a dataset are generated by the same underlying parametric model, differing only in per-view parameters.
    Core MvSR premise stated in Section 2 and Section 4. If violated (e.g., Nikuradse views are an artificial partition of a continuous roughness variable), the benchmark measures something different from the intended multi-view setting.
  • ad hoc to paper Reduced MSE with worst-fitness aggregation is a valid measure of model quality and simplicity.
    Defined in Eq. 4.1 and used throughout. Depends on a consistent definition of nparams, which is problematic for reused parameters and shared constants; test loss is computed with nparams = 0.
  • ad hoc to paper MSE* below 5e-3 indicates a good model.
    Stated in Section 5(a) as a general heuristic without justification from data or theory.
  • domain assumption The four implementations, with mostly default hyperparameters, represent a fair enough comparison for the conclusions drawn.
    The authors acknowledge 'It is rigorously impossible to fairly compare the different MvSR implementations', yet proceed to draw comparative conclusions about which produce good models more frequently.

reviewed 2026-08-05 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Exploring Multi-view Symbolic Regression methods in physical sciences." pith.science (2026). https://pith.science/paper/ZXDTYNKF

@misc{pith2026250910500,
  author       = {Pith},
  title        = {Pith review of: Exploring Multi-view Symbolic Regression methods in physical sciences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXDTYNKF}},
  note         = {Machine review of arXiv:2509.10500}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Describing the world behavior through mathematical functions help scientists to achieve a better understanding of the inner mechanisms of different phenomena. Traditionally, this is done by deriving new equations from first principles and careful observations. A modern alternative is to automate part of this process with symbolic regression (SR). The SR algorithms search for a function that adequately fits the observed data while trying to enforce sparsity, in the hopes of generating an interpretable equation. A particularly interesting extension to these algorithms is the Multi-view Symbolic Regression (MvSR). It searches for a parametric function capable of describing multiple datasets generated by the same phenomena, which helps to mitigate the common problems of overfitting and data scarcity. Recently, multiple implementations added support to MvSR with small differences between them. In this paper, we test and compare MvSR as supported in Operon, PySR, phy-SO, and eggp, in different real-world datasets. We show that they all often achieve good accuracy while proposing solutions with only few free parameters. However, we find that certain features enable a more frequent generation of better models. We conclude by providing guidelines for future MvSR developments.

Figures

Figures reproduced from arXiv: 2509.10500 by Etienne Russeil, Fabr\'icio Olivetti de Fran\c{c}a, Guillaume Moinard, Konstantin Malanchev, Maxime Cherrey.

Figure 1
Figure 1. Figure 1: Cumulative distribution of the reduced MSE aggregated using a worst-fitness function over all datasets and all hyperparameter configurations. The left panel shows the score on training sample. The right panel shows the score on the testing sample. The distributions are clipped such that every value above is displayed in the 5 × 10−2 bin. The medians of the distributions are showed in dashed lines. Medians … view at source ↗
Figure 2
Figure 2. Figure 2: Trade-off plot between the average number of nodes of the most accurate model and the MSE on the test set. The ideal solution should be at the bottom left part of the plot. For readability we display only MSEtest values below 0.025. The Michaelis–Menten dataset contains only a limited number of data points; therefore, in this case, the reported MSEtest metric is not a reliable indicator of performance. 0.0… view at source ↗
Figure 3
Figure 3. Figure 3: Data fit of the supernovae dataset using the lowest reduced MSE solution from each MvSR implementation. For visual clarity, only 3 views are displayed. efficiency of the different methods. More specifically, PySR was substantially slowed down and reached the 30 minutes maximum runtime imposed per experiment. Michaelis-Menten dataset [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Data fit of the galaxy dataset using the lowest reduced MSE solution from each MvSR implementation. For visual clarity, only 5 views are displayed. Operon’s solution includes 5 parameters where other methods were limited to 4. 0.00 0.25 0.50 0.75 1.00 PySR MSEtrain = 0.0003 MSEtest = 0.0173 eggp MSEtrain = 0.0003 MSEtest = 0.0027 0 250 500 750 1000 1250 0.00 0.25 0.50 0.75 1.00 -SO MSEtrain = 0.0015 MSEtes… view at source ↗
Figure 5
Figure 5. Figure 5: Data fit of the Michaelis-Menten dataset using the lowest MSEtrain solution from each MvSR implementation. describes this asymptotic behavior with 2 parameters, and 2 constants. Consequently it is a simple solution, even if its mathematical form is very intricate. The fact that ϕ-SO constructed an efficient solution using a single constant repeated twice confirms the utility of repeated constants, that wou… view at source ↗
Figure 6
Figure 6. Figure 6: Data fit of the Nikuradse dataset using the lowest reduced MSE solution from each MvSR implementation. 0 5 10 15 PySR MSEtrain = 0.0005 MSEtest = 0.0014 eggp MSEtrain = 0.0007 MSEtest = 0.0016 0 2 4 6 8 10 0 5 10 15 -SO MSEtrain = 0.0027 MSEtest = 0.0056 0 2 4 6 8 10 operon MSEtrain = 0.0113 MSEtest = 0.0212 log(order) log(n) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Data fit of the network dataset using the lowest reduced MSE solution from each MvSR implementation. For visual clarity, only 5 views are displayed. while fitting the data very accurately. Network degrees distribution [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

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

  1. Learning the Universe: The Structure of Dust Attenuation Curves in Galaxy Simulations

    astro-ph.GA 2026-06 unverdicted novelty 7.0

    Four parameters suffice to describe dust attenuation curve diversity in TNG simulations, yielding a new symbolic-regression model that recovers curves and fluxes better than existing parameterizations while linking pa...

  2. Symbolic Regression for Shared Expressions: Introducing Partial Parameter Sharing

    cs.LG 2026-01 conditional novelty 6.0

    Introduces partially-shared parameters for symbolic regression with multiple categorical variables, matching prior fit quality on a supernovae dataset with fewer parameters.

Reference graph

Works this paper leans on

35 extracted references · 21 canonical work pages · cited by 2 Pith papers · 3 internal anchors

  1. [1]

    2020 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. In Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion GECCO ’20 p. 1562–1570 New York, NY, USA. Association for Computing Machinery. (10.1145/3377929.3398099)

    work page arXiv 2020

  2. [2]

    2023 Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

    Cranmer M. 2023 Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl. arXiv e-prints p. arXiv:2305.01582. (10.48550/arXiv.2305.01582)

    Pith/arXiv arXiv doi:10.48550/arxiv.2305.01582 2023

  3. [4]

    2021 Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients

    Petersen BK, Larma ML, Mundhenk TN, Santiago CP , Kim SK, Kim JT. 2021 Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. In International Conference on Learning Representations

    work page 2021

  4. [5]

    2023 Deep Symbolic Regression for Physics Guided by Units Constraints: Toward the Automated Discovery of Physical Laws

    Tenachi W, Ibata R, Diakogiannis FI. 2023 Deep Symbolic Regression for Physics Guided by Units Constraints: Toward the Automated Discovery of Physical Laws. apj 959, 99. (10.3847/1538-4357/ad014c)

    doi:10.3847/1538-4357/ad014c 2023

  5. [6]

    McConaghy T. 2011 pp. 235–260. In FFX: Fast, Scalable, Deterministic Symbolic Regression Technology, pp. 235–260. New York, NY: Springer New York

    work page 2011

  6. [7]

    2022 Exhaustive Symbolic Regression

    Bartlett DJ, Desmond H, Ferreira PG. 2022 Exhaustive Symbolic Regression. arXiv e-prints p. arXiv:2211.11461. (10.48550/arXiv.2211.11461)

    Pith/arXiv arXiv doi:10.48550/arxiv.2211.11461 2022

  7. [8]

    2024 syren-new: Precise formulae for the linear and nonlinear matter power spectra with massive neutrinos and dynamical dark energy.arXiv e-prints p

    Sui C, Bartlett DJ, Pandey S, Desmond H, Ferreira PG, Wandelt BD. 2024 syren-new: Precise formulae for the linear and nonlinear matter power spectra with massive neutrinos and dynamical dark energy.arXiv e-prints p. arXiv:2410.14623. (10.48550/arXiv.2410.14623)

    Pith/arXiv arXiv doi:10.48550/arxiv.2410.14623 2024

  8. [9]

    2019 Fast, accurate, and transferable many-body interatomic potentials by symbolic regression

    Hernandez A, Balasubramanian A, Yuan F, Mason SAM, Mueller T. 2019 Fast, accurate, and transferable many-body interatomic potentials by symbolic regression. npj Computational Materials 5, 112. (10.1038/s41524-019-0249-1)

    work page doi:10.1038/s41524-019-0249-1 2019

  9. [10]

    2023 A Flexible Symbolic Regression Method for Constructing Interpretable Clinical Prediction Models

    La Cava WG, Lee PC, Ajmal I, Ding X, Solanki P , Cohen JB, Moore JH, Herman DS. 2023 A Flexible Symbolic Regression Method for Constructing Interpretable Clinical Prediction Models. npj Digital Medicine 6, 1–14. (10.1038/s41746-023-00833-8)

    doi:10.1038/s41746-023-00833-8 2023

  10. [11]

    2023 Understanding conflict origin and dynamics on Twitter: A real-time detection system.Expert Systems with Applications 212, 118748

    de França FO, di Genova DVB, Penteado CLC, Kamienski CA. 2023 Understanding conflict origin and dynamics on Twitter: A real-time detection system.Expert Systems with Applications 212, 118748

    work page 2023

  11. [12]

    2024 Data-Driven Equation Discovery of a Cloud Cover Parameterization

    Grundner A, Beucler T, Gentine P , Eyring V . 2024 Data-Driven Equation Discovery of a Cloud Cover Parameterization. Journal of Advances in Modeling Earth Systems 16, e2023MS003763. (10.1029/2023MS00376310.22541/essoar.168182254.49726852/v1)

    work page arXiv 2024

  12. [13]

    Multi-View Symbolic Regression

    Russeil E, Olivetti de França F, Malanchev K, Burlacu B, Ishida EEO, Leroux M, Michelin C, Moinard G, Gangler E. 2024 Multi-View Symbolic Regression. arXiv e-prints p. arXiv:2402.04298. (10.48550/arXiv.2402.04298)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2402.04298 2024

  13. [14]

    2024 Class Symbolic Regression: Gotta Fit ’Em All

    Tenachi W, Ibata R, François TL, Diakogiannis FI. 2024 Class Symbolic Regression: Gotta Fit ’Em All. apjl 969, L26. (10.3847/2041-8213/ad5970)

    doi:10.3847/2041-8213/ad5970 2024

  14. [15]

    2024 Machine learning mathematical models for incidence estimation during pandemics

    Fajardo-Fontiveros O, Mattei M, Burgio G, Granell C, Gómez S, Arenas A, Sales-Pardo M, Guimerà R. 2024 Machine learning mathematical models for incidence estimation during pandemics. PLoS Computational Biology 20, e1012687. (10.1371/journal.pcbi.1012687)

    work page doi:10.1371/journal.pcbi.1012687 2024

  15. [16]

    2018 Predicting friction system performance with symbolic regression and genetic programming with factor variables

    Kronberger G, Kommenda M, Promberger A, Nickel F. 2018 Predicting friction system performance with symbolic regression and genetic programming with factor variables. In Proceedings of the Genetic and Evolutionary Computation Conference pp. 1278–1285. 15royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 0000000

    work page 2018

  16. [17]

    2021 Contemporary Symbolic Regression Methods and their Relative Performance

    La Cava W, Orzechowski P , Burlacu B, Olivetti de França F, Virgolin M, Jin Y, Kommenda M, Moore JH. 2021 Contemporary Symbolic Regression Methods and their Relative Performance. arXiv e-prints p. arXiv:2107.14351. (10.48550/arXiv.2107.14351)

    Pith/arXiv arXiv doi:10.48550/arxiv.2107.14351 2021

  17. [18]

    2024 SRBench++: Principled benchmarking of symbolic regression with domain-expert interpretation

    Kommenda M, Majumder M, Cranmer M, Espada G, Ingelse L, Fonseca A, Landajuela M, Petersen B et al.. 2024 SRBench++: Principled benchmarking of symbolic regression with domain-expert interpretation. IEEE transactions on evolutionary computation

    work page 2024

  18. [19]

    2025 Call for Action: towards the next generation of symbolic regression benchmark

    Aldeia GSI, Zhang H, Bomarito G, Cranmer M, Fonseca A, Burlacu B, La Cava WG, de França FO. 2025 Call for Action: towards the next generation of symbolic regression benchmark. In Proceedings of the Genetic and Evolutionary Computation Conference Companion GECCO ’25 New York, NY, USA. Association for Computing Machinery. (10.1145/3712255.3734309)

    arXiv 2025

  19. [20]

    2024 The Inefficiency of Genetic Programming for Symbolic Regression

    Kronberger G, Olivetti de Franca F, Desmond H, Bartlett DJ, Kammerer L. 2024 The Inefficiency of Genetic Programming for Symbolic Regression. In Affenzeller M, Winkler SM, Kononova AV , Trautmann H, Tušar T, Machado P , Bäck T, editors,Parallel Problem Solving from Nature – PPSN XVIII pp. 273–289 Cham. Springer Nature Switzerland

    work page 2024

  20. [21]

    2021 Egg: Fast and extensible equality saturation

    Willsey M, Nandi C, Wang YR, Flatt O, Tatlock Z, Panchekha P . 2021 Egg: Fast and extensible equality saturation. Proceedings of the ACM on Programming Languages 5, 1–29

    work page 2021

  21. [22]

    2022 Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery

    Matsubara Y, Chiba N, Igarashi R, Ushiku Y. 2022 Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery. arXiv e-prints p. arXiv:2206.10540. (10.48550/arXiv.2206.10540)

    Pith/arXiv arXiv doi:10.48550/arxiv.2206.10540 2022

  22. [23]

    2022 Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles

    Olivetti de Franca F, Kronberger G. 2022 Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles. arXiv e-prints p. arXiv:2209.06454. (10.48550/arXiv.2209.06454)

    Pith/arXiv arXiv doi:10.48550/arxiv.2209.06454 2022

  23. [24]

    Dark Matter Halos around Galaxies

    Salucci P , Persic M. 1997 Dark Halos around Galaxies. In Persic M, Salucci P , editors,Dark and Visible Matter in Galaxies and Cosmological Implications vol. 117Astronomical Society of the Pacific Conference Series p. 1. (10.48550/arXiv.astro-ph/9703027)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.astro-ph/9703027 1997

  24. [25]

    2020 Navarro-Frenk-White dark matter profile and the dark halos around disk systems

    Dehghani R, Salucci P , Ghaffarnejad H. 2020 Navarro-Frenk-White dark matter profile and the dark halos around disk systems. aap 643, A161. (10.1051/0004-6361/201937079)

    work page doi:10.1051/0004-6361/201937079 2020

  25. [26]

    2011 The original Michaelis constant: translation of the 1913 Michaelis-Menten paper

    Michaelis L, Menten ML, Johnson KA, Goody RS. 2011 The original Michaelis constant: translation of the 1913 Michaelis-Menten paper. Biochemistry 50, 8264–8269

    work page 2011

  26. [27]

    1933 Strömungsgestze in rauhen Rohren

    Nikuradse J. 1933 Strömungsgestze in rauhen Rohren. Technical Report 361 VDI. (English translation: NACA Tech. Memo. 1292, National Advisory Commission for Aeronautics, Washington D.C., 1950)

    work page 1933

  27. [28]

    2020 Bayesian Machine Scientist to Compare Data Collapses for the Nikuradse Dataset

    Reichardt I, Pallarès J, Sales-Pardo M, Guimerà R. 2020 Bayesian Machine Scientist to Compare Data Collapses for the Nikuradse Dataset. Phys. Rev. Lett. 124, 084503. (10.1103/PhysRevLett.124.084503)

    doi:10.1103/physrevlett.124.084503 2020

  28. [29]

    2024 The Inefficiency of Genetic Programming for Symbolic Regression – Extended Version

    Kronberger G, Olivetti de Franca F, Desmond H, Bartlett DJ, Kammerer L. 2024 The Inefficiency of Genetic Programming for Symbolic Regression – Extended Version. arXiv e-prints p. arXiv:2404.17292. (10.48550/arXiv.2404.17292)

    doi:10.48550/arxiv.2404.17292 2024

  29. [30]

    2024 A Comparison of Recent Algorithms for Symbolic Regression to Genetic Programming

    Radwan YA, Kronberger G, Winkler S. 2024 A Comparison of Recent Algorithms for Symbolic Regression to Genetic Programming

    work page 2024

  30. [31]

    1999 Emergence of scaling in random networks

    Barabási AL, Albert R. 1999 Emergence of scaling in random networks. science 286, 509–512

    work page 1999

  31. [32]

    2009 Power-law distributions in empirical data

    Clauset A, Shalizi CR, Newman ME. 2009 Power-law distributions in empirical data. SIAM review 51, 661–703

    work page 2009

  32. [33]

    Handbook of Network Analysis [KONECT -- the Koblenz Network Collection]

    Kunegis J. 2014 Handbook of Network Analysis [KONECT – the Koblenz Network Collection]. arXiv e-prints p. arXiv:1402.5500. (10.48550/arXiv.1402.5500)

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1402.5500 2014

  33. [34]

    2009 The core-collapse rate from the Supernova Legacy Survey

    Bazin G, Palanque-Delabrouille N, Rich J, Ruhlmann-Kleider V , Aubourg E, Le Guillou L, Astier P , Balland C, Basa S, Carlberg RG, Conley A, Fouchez D, Guy J, Hardin D, Hook IM, Howell DA, Pain R, Perrett K, Pritchet CJ, Regnault N, Sullivan M, Antilogus P , Arsenijevic V , Baumont S, Fabbro S, Le Du J, Lidman C, Mouchet M, Mourão A, Walker ES. 2009 The c...

    doi:10.1051/0004- 2009

  34. [35]

    2019 Supernova Photometric Classification Pipelines Trained on Spectroscopically Classified Supernovae from the Pan- STARRS1 Medium-deep Survey.The Astrophysical Journal 884, 83

    Villar VA, Berger E, Miller G, Chornock R, Rest A, Jones DO, Drout MR, Foley RJ, Kirshner R, Lunnan R, Magnier E, Milisavljevic D, Sanders N, Scolnic D. 2019 Supernova Photometric Classification Pipelines Trained on Spectroscopically Classified Supernovae from the Pan- STARRS1 Medium-deep Survey.The Astrophysical Journal 884, 83. (10.3847/1538-4357/ab418c)

    doi:10.3847/1538-4357/ab418c 2019

  35. [36]

    2025 rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models

    de Franca FO, Kronberger G. 2025 rEGGression: an Interactive and Agnostic Tool for the Exploration of Symbolic Regression Models. In Proceedings of the Genetic and Evolutionary Computation Conference GECCO ’25 New York, NY, USA. Association for Computing Machinery. (10.1145/3712256.3726385)

    work page arXiv 2025

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.