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Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A pre-trained LLM's equation scores, added to the symbolic-regression loss, consistently improve recovery of physical dynamics from noisy data, with richer prompts helping more.

desk verdict LLM-as-scorer inside the SR loss is a genuinely new mechanism, but the experiments don't yet rule out the simpler story that it is just another complexity penalty. read the letter →

arxiv 2509.03036 v1 pith:64J6QDUN submitted 2025-09-03 cs.LG cs.AIcs.IRcs.SC

Bilge Taskin , Wenxiong Xie , Teddy Lazebnik This is my paper · ORCID
classification cs.LGcs.AIcs.IRcs.SC
keywords symbolicregressionphysics-informedlargelanguagemodelsknowledgeintegrationlossfunctionexpressiontreescorenoiserobustnesspromptengineering
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Symbolic regression finds equations from data, but data alone often leads to fits that are numerically good and physically wrong. This paper tries to make the search physically aware without hand-crafted constraints by asking a pre-trained large language model to grade each candidate equation on dimensional correctness, simplicity, and realism, and adding that grade to the regression loss. Across three genetic-programming symbolic-regression algorithms, three language models, and three physical dynamics (free fall, harmonic motion, damped wave), the LLM term consistently lowered error and improved structural recovery of the true equations relative to the same algorithms without the LLM. The paper also reports that the method degrades more gracefully than baselines as Gaussian noise grows, and that prompts carrying more experimental context produce better and more exact reconstructions. If this result holds, a generic language model can act as a reusable physics prior for equation discovery, replacing hand-built constraints.

What carries the argument

The load-bearing object is the LLM score term $c$ and the prompt that produces it. The prompt fixes a role for the model as a scientific-reasoning assistant, defines the three metrics with numerical scales, supplies three few-shot examples, and demands a Python-style list output; the sampling temperature is set to zero so the score is deterministic. This converts an open-ended language model into a repeatable evaluator whose score is combined with data fit and equation size in the symbolic-regression loss, steering the genetic search toward equations that are simultaneously accurate, compact, and physically plausible.

What would settle it

Give the same three tasks to the method with the LLM term replaced by a pure equation-length penalty, or compare the LLM's scores to expert physicists' ratings of the same candidate equations; if either control reproduces or fails to correlate with the reported gains, the physical-knowledge claim is undermined.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a pre-trained LLM can serve as a physics-informed evaluator inside the symbolic-regression loop. The proposed loss is $L=w_1 e + w_2 s + w_3 c$, where $e$ is the mean squared error between predicted and observed values, $s$ is the candidate equation's size in expression-tree nodes, and $c$ is the LLM's penalty score. The LLM is prompted to return three sub-scores, dimensional correctness, simplicity, and realism, on $[0,1]$ scales, and $c$ is defined as $1-(c_1+c_2+c_3)/3$. The paper argues that this single term injects domain knowledge that is normally encoded by specialized formulations or manual feature engineering, and presents experiments in which every LLM-integrated configuration outperforms its no-LLM baseline on fitting error and expression-tree similarity, with the best configurations recovering the ground-truth equation exactly.

Load-bearing premise

The load-bearing premise is that the LLM's zero-shot scores for dimensional consistency, simplicity, and realism measure physical validity and not surface features like equation length or phrasing familiarity, because the loss term can only help if the score rewards physics rather than style.

Editorial extensions

If this is right

  • Every tested LLM-integrated configuration outperformed the same symbolic-regression algorithm without the LLM, so the reported improvement does not depend on a single regression engine.
  • Prompts that supplied variable descriptions, experiment context, or the ground-truth formula improved performance monotonically, and combining all three cues made the recovered expression tree structurally identical to the ground truth in every engine-model pair.
  • Adding the LLM term raises the expression-tree score at every tested noise level from 1% to 5%, with the largest gains visible when both input features and target values are corrupted.
  • The method's sensitivity to noise still grows with the complexity of the true equation, meaning the LLM prior reduces but does not eliminate the difficulty of hard reconstruction problems.
  • Because the LLM term is prompt-driven, it can be applied to new physical domains by changing the context string, with no change to the symbolic-regression engine.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A decisive control experiment would replace the LLM with an automatic penalty that only rewards shorter equations; if the gains persist, the reported improvements come from complexity regularization rather than physical knowledge.
  • The perfect structural scores under prompts that reveal the ground-truth formula are expected if the LLM simply echoes the formula back; the more interesting test is whether the gain survives when the LLM is not shown the answer.
  • If the mechanism is genuine physics knowledge, the benefit should transfer to dynamics underrepresented in the LLM's training corpus; a hold-out test on an unfamiliar physics domain would separate memorization from reasoning.
  • Because the LLM score is sampled at zero temperature, score stability across stochastic samples is untested; sampling at higher temperatures could reveal how reliably the LLM ranks competing equations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper proposes integrating pre-trained large language models into symbolic regression by adding an LLM-based score term to the SR loss function. The method is evaluated on three physical dynamics (free fall, simple harmonic motion, damped wave), three SR algorithms (DEAP, gplearn, PySR), and three LLMs (Mistral, Llama 2, Falcon). The authors report consistent improvements in MAE, MSE, R2, and expression-tree score, investigate eight prompt variants, and test robustness to 1%–5% Gaussian noise. The central claim is that a generic LLM prompt can serve as a reusable physics prior that improves reconstruction and robustness of SR models.

Significance. If the central claim is validated, the proposed approach would be a practically useful and relatively accessible way to inject domain knowledge into symbolic regression, since it avoids handcrafted constraints and specialized formulations. The paper has several strengths: the code and data are publicly released, the LLM sampling temperature is set to zero for reproducibility, the evaluation covers a 3x3x3 grid of components, and the expression-tree score provides a structural similarity metric in addition to numerical fit. However, the current evidence does not yet establish that the reported gains come from physics knowledge encoded in the LLM scores rather than from an implicit complexity penalty or from other confounds. The authors themselves acknowledge in Section 6 that the LLM score was not validated by human experts and that it is unclear whether gains reflect physical realism or technical properties such as equation length. Because this point is load-bearing for the paper's main claim, the manuscript needs additional experiments and reporting before it can be accepted.

major comments (5)
  1. [Eq. (1), Section 3, Section 6] The central claim requires that the LLM score c in Eq. (1) measures physical validity, but the reported definition c := 1 - (c1 + c2 + c3)/3 includes c2, which is explicitly a simplicity score. Expanding Eq. (1) gives L = w1*e + (w2 + w3/3)*s + w3/3*(1 - c1 - c3), so increasing w3 is partly just increasing the complexity penalty. Since the loss weights w1, w2, w3 are never reported, a reader cannot distinguish the proposed knowledge-integration mechanism from a conventional complexity regularizer. The manuscript's own Section 6 limitation concedes this exact concern. The authors should report the weights, provide an ablation that removes the simplicity component (or uses only c1 and c3), and compare against a non-LLM complexity-only baseline with matched penalty strength.
  2. [Table 4, Section 4.2.3, Section 6] The noise robustness analysis does not support the stated claim. Section 6 says that 'the LLM-integrated SR model improves the robustness to high levels of all three noise types compared to the same model without the LLM (baseline),' but Table 4 contains only DEAP, gplearn, and PySR rows and no LLM-free baseline rows. Without those baseline rows, the table shows only that PySR degrades more gracefully than DEAP and gplearn under noise, which is a property of the SR algorithms, not of LLM integration. The authors should add the corresponding no-LLM baseline results for all three SR models under the same noise conditions.
  3. [Table 2, Section 4.2.1] Every entry in Table 2 appears to be a single run of a stochastic symbolic regression algorithm. Since DEAP, gplearn, and PySR all have randomized search components, the reported uniform improvements could be within run-to-run variability. The authors should report multiple independent seeds with means and standard deviations, or otherwise provide a statistical comparison (e.g., paired tests) between the LLM-integrated and baseline configurations.
  4. [Table 3, Section 4.2.2] The prompt-sensitivity experiment contains a circular evaluation. Prompts D, F, G, and H append the ground-truth equation to the LLM prompt, so the resulting expression-tree scores of 1.00 and 'perfect reconstruction' are expected consequences of providing the answer, not evidence that the LLM contributes physics-aware reasoning. These rows cannot be used to support the claim that richer prompts improve knowledge integration. The paper should either exclude the GT-containing prompts from the main evidence or reframe them explicitly as an upper-bound check rather than as a demonstration of the method.
  5. [Table 3 (rows E, F, G, H), Section 4.2.2] In Table 3, the rows for prompts E, F, G, and H are identical within each LLM-SR pair (same MAE, MSE, R2, and tree score). This is surprising because these prompts differ in content, and it is unclear whether this reflects a copy/paste error, a reporting artifact, or a genuine insensitivity of the pipeline to all prompt differences once variable descriptions and experiment descriptions are present. The authors should clarify this in the text and verify the reported values against the actual experimental logs.
minor comments (5)
  1. [Section 4.1.1] The sentence 'the three physical cases are: free fall of a ball, simple harmonic motion, and damped wave—were' contains a grammatical error and should be rewritten.
  2. [Section 4.1.2, Section 6] There are typographical errors: 'locally-depolyable' in Section 4.1.2, 'comparision' in Section 4, and 'DEAP, Falcom' in Section 6 should be corrected.
  3. [Table 1] The hyperparameter table appears inconsistent with the text: Section 4.1.3 states DEAP uses crossover probability 0.6 and mutation probability 0.05, but Table 1 lists Crossover probability '—' for DEAP and Mutation probability 0.01. Please align the table with the text or explain the discrepancy.
  4. [Section 4.3] The expression-tree distance definition includes a coefficient alpha in [0,1] for differing numeric leaf values, but the value of alpha used in the experiments is never reported. Please state the value or explain how it was chosen.
  5. [References] References [113] and [114] appear to refer to the same work (Sharlin and Josephson, 'In Context Learning and Reasoning for Symbolic Regression with Large Language Models') and should be merged or distinguished.

Circularity Check

2 steps flagged · score 4.0 of 10

Two supporting results are circular by construction: GT-containing prompts produce perfect recovery, and the LLM score includes a simplicity subscore that can mimic the baseline complexity penalty; the Prompt-A benchmark itself is not circular.

  1. self definitional [Section 5 (Results), Table 3 discussion; prompt definitions in Section 4.2.2]
    "Importantly, for the cases where the GT equation is provided to the LLM (Cases D, F, G, and H), an expression tree score of 1.00 is obtained, meaning a perfect reconstruction of the GT equation."

    Prompts D, F, G, and H are defined in Section 4.2.2 as including the GT equation (D appends the GT equation; F = B + D; G = C + D; H = B + C + D). The perfect expression-tree score of 1.00 in these cases is therefore tautological: the target equation was written into the LLM's prompt, so the SR search is steered directly to the ground truth. Counting those cases as evidence that 'more informative prompts significantly improve performance' equates the method's output with its own input. The result is forced by construction and cannot support the claim that pre-trained LLMs supply physical knowledge.

  2. other [Section 3, loss function Eq. (1), LLM score definition and prompt; Section 6 limitation]
    "The LLM’s overall score is computed as follows: c := 1− (c1 +c2 +c3)/3. ### METRICS simp . 0 (complex) -> 1 (simple)"

    By construction, the added LLM term c contains c2, a simplicity score. Substituting into Eq. (1), L = w1e + w2s + w3(1 − (c1+c2+c3)/3) = w1e + w2s + w3 − (w3/3)(c1+c2+c3), so a higher c2 lowers the loss, rewarding shorter expressions and duplicating the existing size penalty s. The reported gains over the LLM-free baseline can therefore be fully explained by an enhanced complexity-regularization term rather than by physics knowledge. The paper's Section 6 limitation concedes this: 'it is challenging to gauge whether performance gains were due to genuinely helpful scores in terms of physical realism or the score captures more technical properties, such as the equation's length.' The central evidence for 'physics-informed' knowledge is not separated from this confound by construction.

full rationale

The core benchmark comparison uses Prompt A, which contains no ground-truth equation, so the main claim that LLM integration improves reconstruction is not itself circular: the LLM score is an independent input in that setting. However, two load-bearing pieces of supporting evidence reduce to their own inputs. First, the prompt-sensitivity analysis reports perfect recovery for prompts D, F, G, and H, all of which include the GT equation; that success is tautological. Second, the LLM score c is defined as 1 minus the average of dimensional-correctness, simplicity, and realism subscores, so the added loss term is partially a simplicity regularizer; the paper's own limitation admits that gains may stem from equation length rather than physical realism. The unreported weights w1, w2, w3 and absent LLM-free rows in Table 4 further weaken the separation, but those are experimental-reporting weaknesses rather than circular reductions. Self-citations to [43], [71], and [136] (co-authored by Lazebnik) are present but are not used as load-bearing uniqueness or validation arguments for the central result. Overall, the paper's main Prompt-A result has independent content, but an important subset of its success evidence is circular or confounded, giving a score of 4.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on an unvalidated LLM scoring assumption, an unreported set of loss weights, and the representativeness of three synthetic benchmarks. These are the principal costs a reader pays before the empirical results can be interpreted.

free parameters (3)
  • Loss weights w1, w2, w3 = not reported
    Equation (1) defines L = w1*e + w2*s + w3*c with w1+w2+w3=1, but the actual values used in experiments are never stated. These weights control the trade-off between data fit, complexity, and LLM score, so the central result depends on an unreported choice.
  • Expression-tree leaf mismatch coefficient alpha = not reported
    Section 4.3 defines numeric-leaf mismatch as alpha*|v1-v2| capped at 1 for alpha in [0,1], but the chosen alpha is not given, making tree-score thresholds unreproducible.
  • LLM sub-score aggregation weights = 1/3 each
    The overall LLM score c = 1 - (c1+c2+c3)/3 gives equal weight to dimensional consistency, simplicity, and realism, an arbitrary equality with no stated justification.
assumptions (4)
  • domain assumption The LLM's zero-shot scalar scores for dimensional consistency, simplicity, and realism are accurate enough to guide SR search.
    Section 3 defines the loss with c derived from LLM outputs; the paper later concedes in Section 6 that these scores were never manually or systematically validated.
  • domain assumption The three synthetic dynamics and the data generation procedure are representative cases for physics-informed symbolic regression.
    Section 4.1.1 generates data for free fall, harmonic motion, and damped wave only, yet general claims of consistent improvement are drawn from these cases.
  • standard math Expression-tree distance, with the specified tree comparison rules, is a valid structural similarity metric.
    Section 4.3 uses expression-tree distance to define the tree score, relying on standard tree edit distance concepts.
  • domain assumption The genetic-programming baselines with chosen hyperparameters are adequate comparators for improvement.
    Section 4.1.3 fixes population size, generations, and mutation rates without tuning, so baseline strength is assumed rather than established.

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Pith. "Pith review of Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models." pith.science (2026). https://pith.science/paper/64J6QDUN

@misc{pith2026250903036,
  author       = {Pith},
  title        = {Pith review of: Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64J6QDUN}},
  note         = {Machine review of arXiv:2509.03036}
}
read the original abstract

Symbolic regression (SR) has emerged as a powerful tool for automated scientific discovery, enabling the derivation of governing equations from experimental data. A growing body of work illustrates the promise of integrating domain knowledge into the SR to improve the discovered equation's generality and usefulness. Physics-informed SR (PiSR) addresses this by incorporating domain knowledge, but current methods often require specialized formulations and manual feature engineering, limiting their adaptability only to domain experts. In this study, we leverage pre-trained Large Language Models (LLMs) to facilitate knowledge integration in PiSR. By harnessing the contextual understanding of LLMs trained on vast scientific literature, we aim to automate the incorporation of domain knowledge, reducing the need for manual intervention and making the process more accessible to a broader range of scientific problems. Namely, the LLM is integrated into the SR's loss function, adding a term of the LLM's evaluation of the SR's produced equation. We extensively evaluate our method using three SR algorithms (DEAP, gplearn, and PySR) and three pre-trained LLMs (Falcon, Mistral, and LLama 2) across three physical dynamics (dropping ball, simple harmonic motion, and electromagnetic wave). The results demonstrate that LLM integration consistently improves the reconstruction of physical dynamics from data, enhancing the robustness of SR models to noise and complexity. We further explore the impact of prompt engineering, finding that more informative prompts significantly improve performance.

Figures

Figures reproduced from arXiv: 2509.03036 by the authors.

Figure 1
Figure 1. A schematic view of the Large Language Model-based Knowledge Integration into the Symbolic Re [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. A schematic view of the experimental setup. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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Works this paper leans on

144 extracted references · 40 canonical work pages

  1. [1]

    Newton and the relation of mathematics to natural philosophy

    J. W. Garrison. “Newton and the relation of mathematics to natural philosophy”. In: Journal of the History of Ideas 48.4 (1987), pp. 609–627

    1987

  2. [2]

    Guicciardini

    N. Guicciardini. Isaac Newton on mathematical certainty and method. 4. Mit Press, 2009

    2009

  3. [3]

    Newton, Maclaurin, and the authority of Mathematics

    J. V . Grabiner. “Newton, Maclaurin, and the authority of Mathematics”. In: The American Mathematical Monthly 111.10 (2004), pp. 841–852

    2004

  4. [4]

    Revisiting the mathematisation thesis: Galileo, Descartes, Newton, and the Language of Na- ture

    L. Kvasz. “Revisiting the mathematisation thesis: Galileo, Descartes, Newton, and the Language of Na- ture”. In: International Studies in the Philosophy of Science 30.4 (2016), pp. 399–406

    2016

  5. [5]

    R. K. Merton. The sociology of science: Theoretical and empirical investigations . University of Chicago press, 1973

    1973

  6. [6]

    D. J. D. S. Price. Little science, big science. Columbia university press, 1963

    1963

  7. [7]

    Using math in physics: Overview

    E. F. Redish. “Using math in physics: Overview”. In: The Physics Teacher 59.5 (2021), pp. 314–318

    2021

  8. [8]

    E. K. Yeargers, J. V . Herod, and R. W. Shonkweiler. An introduction to the mathematics of biology: with computer algebra models. Springer Science & Business Media, 2013

    2013

Show all 144 references
  1. [9]

    The mathematization of economic theory

    G. Debreu. “The mathematization of economic theory”. In: The American Economic Review 81.1 (1991), pp. 1–7

    1991

  2. [10]

    J. C. Maxwell. Electricity and magnetism. V ol. 2. Dover New York, 1954

    1954

  3. [11]

    The general theory of relativity

    A. Einstein. “The general theory of relativity”. In: The meaning of relativity. Springer, 1922, pp. 54–75

    1922

  4. [12]

    What did mathematics do to physics?

    Y . Gingras. “What did mathematics do to physics?” In: History of science 39.4 (2001), pp. 383–416

    2001

  5. [13]

    N. R. Council, D. on Engineering, P. Sciences, B. on Mathematical Sciences, T. Applications, C. on the Mathematical Sciences in, and 2025. Fueling Innovation and Discovery: The Mathematical Sciences in the 21st Century. National Academies Press, 2012

    2025

  6. [14]

    G. Polya. Mathematical discovery, 1962. John Wiley & Sons, 1962. Draft: September 8, 2025 18

    1962

  7. [15]

    Fostering hooks and shifts: Tutorial tactics for guided mathematical discovery

    D. Abrahamson, J. Guti ´errez, T. Charoenying, A. Negrete, and E. Bumbacher. “Fostering hooks and shifts: Tutorial tactics for guided mathematical discovery”. In: Technology, Knowledge and Learning 17 (2012), pp. 61–86

    2012

  8. [16]

    Naturalising mathematics? a wittgensteinian perspective

    J. Stam, M. Stokhof, and M. Van Lambalgen. “Naturalising mathematics? a wittgensteinian perspective”. In: Philosophies 7.4 (2022), p. 85

    2022

  9. [17]

    How a scientific discovery is made: A case history

    G. Holton, H. Chang, and E. Jurkowitz. “How a scientific discovery is made: A case history”. In: American Scientist 84.4 (1996), pp. 364–375

    1996

  10. [18]

    T. Nickles. Scientific discovery, logic, and rationality. V ol. 56. Springer Science & Business Media, 2012

    2012

  11. [19]

    Stevens, V

    R. Stevens, V . Taylor, J. Nichols, A. B. Maccabe, K. Yelick, and D. Brown. Ai for science: Report on the department of energy (doe) town halls on artificial intelligence (ai) for science . Tech. rep. Argonne National Lab.(ANL), Argonne, IL (United States), 2020

    2020

  12. [20]

    AI and science: what 1,600 researchers think

    R. Van Noorden and J. M. Perkel. “AI and science: what 1,600 researchers think”. In: Nature 621.7980 (2023), pp. 672–675

    2023

  13. [21]

    On scientific understanding with artificial intelligence

    M. Krenn et al. “On scientific understanding with artificial intelligence”. In: Nature Reviews Physics 4.12 (2022), pp. 761–769

    2022

  14. [22]

    AI and the transformation of social science research

    I. Grossmann, M. Feinberg, D. C. Parker, N. A. Christakis, P. E. Tetlock, and W. A. Cunningham. “AI and the transformation of social science research”. In: Science 380.6650 (2023), pp. 1108–1109

    2023

  15. [23]

    From big data analysis to personalized medicine for all: challenges and opportunities

    A. Alyass, M. Turcotte, and D. Meyre. “From big data analysis to personalized medicine for all: challenges and opportunities”. In: BMC medical genomics 8 (2015), pp. 1–12

    2015

  16. [24]

    Economics in the age of big data

    L. Einav and J. Levin. “Economics in the age of big data”. In: Science 346.6210 (2014), p. 1243089

    2014

  17. [25]

    AI meets physics: a comprehensive survey

    L. Jiao et al. “AI meets physics: a comprehensive survey”. In: Artificial Intelligence Review 57.9 (2024), p. 256

    2024

  18. [26]

    The artificial intelligence revolution in digital finance in Saudi Arabia: A comprehensive review and proposed framework

    H. H. Al-Baity. “The artificial intelligence revolution in digital finance in Saudi Arabia: A comprehensive review and proposed framework”. In: Sustainability 15.18 (2023), p. 13725

    2023

  19. [27]

    Fifth revolution: Applied AI & human intelligence with cyber physical systems

    P. Pathak, P. R. Pal, M. Shrivastava, and P. Ora. “Fifth revolution: Applied AI & human intelligence with cyber physical systems”. In: International Journal of Engineering and Advanced Technology 8.3 (2019), pp. 23–27

    2019

  20. [28]

    A survey on explainable artificial intelligence (xai): Toward medical xai

    E. Tjoa and C. Guan. “A survey on explainable artificial intelligence (xai): Toward medical xai”. In: IEEE transactions on neural networks and learning systems 32.11 (2020), pp. 4793–4813

    2020

  21. [29]

    Opening the black box: the promise and limitations of explainable machine learning in cardiology

    J. Petch, S. Di, and W. Nelson. “Opening the black box: the promise and limitations of explainable machine learning in cardiology”. In: Canadian Journal of Cardiology 38.2 (2022), pp. 204–213

    2022

  22. [30]

    Opening the black box: interpretable machine learning for geneti- cists

    C. B. Azodi, J. Tang, and S.-H. Shiu. “Opening the black box: interpretable machine learning for geneti- cists”. In: Trends in genetics 36.6 (2020), pp. 442–455

    2020

  23. [31]

    Meaningful explanations of black box AI decision systems

    D. Pedreschi, F. Giannotti, R. Guidotti, A. Monreale, S. Ruggieri, and F. Turini. “Meaningful explanations of black box AI decision systems”. In: Proceedings of the AAAI conference on artificial intelligence . V ol. 33. 01. 2019, pp. 9780–9784. Draft: September 8, 2025 19

    2019

  24. [32]

    Causability and explainability of artificial intelligence in medicine

    A. Holzinger, G. Langs, H. Denk, K. Zatloukal, and H. M ¨uller. “Causability and explainability of artificial intelligence in medicine”. In: Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery9.4 (2019), e1312

    2019

  25. [33]

    Interpretable scientific discovery with symbolic regression: a review

    N. Makke and S. Chawla. “Interpretable scientific discovery with symbolic regression: a review”. In: Artificial Intelligence Review 57.1 (2024), p. 2

    2024

  26. [34]

    Kronberger, B

    G. Kronberger, B. Burlacu, M. Kommenda, S. M. Winkler, and M. Affenzeller. Symbolic Regression. CRC Press, 2024

    2024

  27. [35]

    Symbolic regression in materials science

    Y . Wang, N. Wagner, and J. M. Rondinelli. “Symbolic regression in materials science”. In: MRS commu- nications 9.3 (2019), pp. 793–805

    2019

  28. [36]

    Artificial intelligence in physical sciences: Symbolic re- gression trends and perspectives

    D. Angelis, F. Sofos, and T. E. Karakasidis. “Artificial intelligence in physical sciences: Symbolic re- gression trends and perspectives”. In: Archives of Computational Methods in Engineering 30.6 (2023), pp. 3845–3865

    2023

  29. [37]

    AI Feynman: A physics-inspired method for symbolic regression

    S.-M. Udrescu and M. Tegmark. “AI Feynman: A physics-inspired method for symbolic regression”. In: Science advances 6.16 (2020), eaay2631

    2020

  30. [38]

    Distilling Free-Form Natural Laws from Experimental Data

    M. Schmidt and H. Lipson. “Distilling Free-Form Natural Laws from Experimental Data”. In: Science 324.5923 (2009), pp. 81–85

    2009

  31. [39]

    Automated Rediscovery of the Maxwell Equations

    M. Stalzer and C. Ju. “Automated Rediscovery of the Maxwell Equations”. In: Applied Sciences 9.14 (2019), p. 2899. DOI: 10.3390/app9142899

    2019 doi

  32. [40]

    Discovering symbolic models from deep learn- ing with inductive biases

    M. Cranmer, R. Xu, P. Battaglia, D. N. Spergel, and S. Ho. “Discovering symbolic models from deep learn- ing with inductive biases”. In: Nature Communications 11.1 (2020), p. 5354. DOI: 10.1038/s41467- 020-18020-0

    2020 doi

  33. [42]

    Discovering governing equations from data by sparse iden- tification of nonlinear dynamical systems

    S. L. Brunton, J. L. Proctor, and J. N. Kutz. “Discovering governing equations from data by sparse iden- tification of nonlinear dynamical systems”. In: Proceedings of the National Academy of Sciences 113.15 (2016), pp. 3932–3937. DOI: 10.1073/pnas.1517384113

    2016 doi

  34. [44]

    Multi-objective symbolic regression for physics-aware dynamic modeling

    J. Kubal ´ık, E. Derner, and R. Babu ˇska. “Multi-objective symbolic regression for physics-aware dynamic modeling”. In: Expert Systems with Applications 182 (2021), p. 115210

    2021

  35. [45]

    Toward physically plausible data-driven models: A novel neural network approach to symbolic regression

    J. Kubal ´ık, E. Derner, and R. Babu ˇska. “Toward physically plausible data-driven models: A novel neural network approach to symbolic regression”. In: IEEE Access 11 (2023), pp. 61481–61501

    2023

  36. [46]

    Effectiveness of generative artificial intelligence for scientific content analysis

    M. Platt and D. Platt. “Effectiveness of generative artificial intelligence for scientific content analysis”. In: 2023 IEEE 17th International Conference on Application of Information and Communication Technolo- gies (AICT). IEEE. 2023, pp. 1–4. Draft: September 8, 2025 20

    2023

  37. [47]

    Theory-guided data science: a new paradigm for scientific discovery from data

    A. Karpatne, G. Atluri, J. H. Faghmous, M. Steinbach, A. Banerjee, A. R. Ganguly, S. Shekhar, N. Sama- tova, and V . Kumar. “Theory-guided data science: a new paradigm for scientific discovery from data”. In: IEEE Transactions on Knowledge and Data Engineering 29 (10 2017), pp...

    2017

  38. [48]

    Automated discovery systems, part 2: new developments, current issues, and philosophical lessons in machine learning and data science

    P. Giza. “Automated discovery systems, part 2: new developments, current issues, and philosophical lessons in machine learning and data science”. In: Philosophy Compass 17 (1 2021). DOI: 10.1111/ phc3.12802

    2021

  39. [49]

    The rise of self-driving labs in chemical and materials sciences

    M. Abolhasani and E. Kumacheva. “The rise of self-driving labs in chemical and materials sciences”. In: Nature Synthesis 2 (6 2023), pp. 483–492. DOI: 10.1038/s44160-022-00231-0

    2023 doi

  40. [50]

    Towards robot scientists for autonomous scientific discovery

    A. C. Sparkes et al. “Towards robot scientists for autonomous scientific discovery”. In: Automated Exper- imentation 2 (1 2010), p. 1. DOI: 10.1186/1759-4499-2-1

    2010 doi

  41. [51]

    Automatic discovery and optimization of chemical processes

    C. Houben and A. Lapkin. “Automatic discovery and optimization of chemical processes”. In: Current Opinion in Chemical Engineering 9 (2015), pp. 1–7. DOI: 10.1016/j.coche.2015.07.001

    2015 doi

  42. [52]

    On the issue of automation of processes during scientific research

    E. Zubrilina and O. S. Babenko. “On the issue of automation of processes during scientific research”. In: E3S Web of Conferences493 (2024), p. 01018. DOI: 10.1051/e3sconf/202449301018

    2024

  43. [53]

    Computational scientific discovery in psychology

    L. Bartlett, A. Pirrone, N. Javed, and F. Gobet. “Computational scientific discovery in psychology”. In: Perspectives on Psychological Science18 (1 2022), pp. 178–189.DOI: 10.1177/17456916221091833

    2022 doi

  44. [54]

    Kramer, M

    S. Kramer, M. Cerrato, S. D ˇzeroski, and R. King. Automated Scientific Discovery: From Equation Dis- covery to Autonomous Discovery Systems . 2023. arXiv: 2305 . 02251 [cs.AI]. URL: https : / / arxiv.org/abs/2305.02251

    2023 arXiv

  45. [55]

    BACON: A Production System That Discovers Empirical Laws

    P. Langley. “BACON: A Production System That Discovers Empirical Laws.” In: IJCAI. Citeseer. 1977, p. 344

    1977

  46. [56]

    P. Langley. Scientific discovery: Computational explorations of the creative processes. MIT press, 1987

    1987

  47. [57]

    Discovering dynamics

    S. D ˇzeroski and L. Todorovski. “Discovering dynamics”. In: Proc. tenth international conference on ma- chine learning. 1993, pp. 97–103

    1993

  48. [58]

    Genetic programming as a means for programming computers by natural selection

    J. R. Koza. “Genetic programming as a means for programming computers by natural selection”. In: Statistics and computing 4 (1994), pp. 87–112

    1994

  49. [59]

    Discovering dynamics with genetic programming

    S. D ˇzeroski and I. Petrovski. “Discovering dynamics with genetic programming”. In: Machine Learning: ECML-94: European Conference on Machine Learning Catania, Italy, April 6–8, 1994 Proceedings 7 . Springer. 1994, pp. 347–350

    1994

  50. [60]

    Functional genomic hypothesis generation and experimentation by a robot scientist

    R. D. King, K. E. Whelan, F. M. Jones, P. G. K. Reiser, C. H. Bryant, S. Muggleton, D. B. Kell, and S. G. Oliver. “Functional genomic hypothesis generation and experimentation by a robot scientist”. In: Nature 427 (6971 2004), pp. 247–252. DOI: 10.1038/nature02236

    2004 doi

  51. [61]

    The automation of science

    R. D. King et al. “The automation of science”. In: Science 324 (5923 2009), pp. 85–89. DOI: 10.1126/ science.1165620

    2009

  52. [62]

    Automated discovery systems, part 1: historical origins, main research programs, and method- ological foundations

    P. Giza. “Automated discovery systems, part 1: historical origins, main research programs, and method- ological foundations”. In: Philosophy Compass 17 (1 2021). DOI: 10.1111/phc3.12800. Draft: September 8, 2025 21

    2021 doi

  53. [63]

    Toward an artificial intelligence physicist for unsupervised learning

    T. Wu and M. Tegmark. “Toward an artificial intelligence physicist for unsupervised learning”. In:Physical Review E 100 (3 2019). DOI: 10.1103/physreve.100.033311

    2019 doi

  54. [64]

    Modeling hierarchy using symbolic regression

    I. Icke and J. Bongard. “Modeling hierarchy using symbolic regression”. In: 2013 IEEE Congress on Evolutionary Computation (2013), pp. 2980–2987. DOI: 10.1109/cec.2013.6557932

    2013

  55. [65]

    Symbolic regression by exhaustive search: reducing the search space using syntactical constraints and efficient semantic structure deduplication

    L. Kammerer, G. Kronberger, B. Burlacu, S. Winkler, M. Kommenda, and M. Affenzeller. “Symbolic regression by exhaustive search: reducing the search space using syntactical constraints and efficient semantic structure deduplication”. In: Genetic and Evolutionary Computation (20...

    2020 doi

  56. [66]

    Symbolicgpt: a generative transformer model for symbolic regression

    M. Valipour, B. You, M. Panju, and A. Ghodsi. “Symbolicgpt: a generative transformer model for symbolic regression”. In: arXiv preprint arXiv:2106.14131 (2021). DOI: 10.48550/arxiv.2106.14131

    2021 doi arXiv

  57. [67]

    Interpretable scientific discovery with symbolic regression: a review

    N. Makke and S. Chawla. “Interpretable scientific discovery with symbolic regression: a review”. In: Artificial Intelligence Review 57 (1 2024). DOI: 10.1007/s10462-023-10622-0

    2024 doi

  58. [68]

    Block building programming for symbolic regression

    C. Chen, C. Luo, and Z. Jiang. “Block building programming for symbolic regression”. In: Neurocomput- ing 275 (2018), pp. 1973–1980. DOI: 10.1016/j.neucom.2017.10.047

    2018 doi

  59. [69]

    Bayesian symbolic regression

    Y . Jin, W. Fu, J. Kang, J. Guo, and J. Guo. “Bayesian symbolic regression”. In: (2019). DOI: 10.48550/ arxiv.1910.08892

    2019 doi arXiv

  60. [71]

    A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge

    L. S. Keren, A. Liberzon, and T. Lazebnik. “A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge”. In:Scientific Reports 13 (1 2023). DOI: 10.1038/s41598-023-28328-2

    2023 doi

  61. [72]

    R. Riolo. Genetic programming theory and practice X. Springer, 2013

    2013

  62. [73]

    The science of brute force

    M. J. Heule and O. Kullmann. “The science of brute force”. In: Communications of the ACM 60.8 (2017), pp. 70–79

    2017

  63. [74]

    Discovering governing equations from data by sparse iden- tification of nonlinear dynamical systems

    S. L. Brunton, J. L. Proctor, and J. N. Kutz. “Discovering governing equations from data by sparse iden- tification of nonlinear dynamical systems”. In: Proceedings of the national academy of sciences 113.15 (2016), pp. 3932–3937

    2016

  64. [75]

    PySINDy: A comprehensive Python package for robust sparse system identifica- tion

    A. A. Kaptanoglu et al. “PySINDy: A comprehensive Python package for robust sparse system identifica- tion”. In: arXiv preprint arXiv:2111.08481 (2021)

    2021 arXiv

  65. [76]

    Contemporary Symbolic Regression Methods and Their Relative Performance

    W. La Cava, P. Orzechowski, B. Burlacu, F. O. de Franc ¸a, M. Virgolin, Y . Jin, M. Kommenda, and J. H. Moore. “Contemporary Symbolic Regression Methods and Their Relative Performance”. In: Thirty-fifth Conference on Neural Information Processing Systems (NeurIPS 2021) Dataset...

    2021

  66. [77]

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

    B. K. Petersen, M. Landajuela, T. N. Mundhenk, C. Santiago, S. K. Kim, and J. T. Kim. “Deep symbolic regression: recovering mathematical expressions from data via risk-seeking policy gradients”. In: (2019). DOI: 10.48550/arxiv.1912.04871

    2019 doi arXiv

  67. [78]

    Where are we now?

    P. Orzechowski, W. L. Cava, and J. H. Moore. “Where are we now?” In: Proceedings of the Genetic and Evolutionary Computation Conference (2018). DOI: 10.1145/3205455.3205539. Draft: September 8, 2025 22

    2018

  68. [79]

    Improving model-based genetic pro- gramming for symbolic regression of small expressions

    M. Virgolin, T. Alderliesten, C. Witteveen, and P. A. N. Bosman. “Improving model-based genetic pro- gramming for symbolic regression of small expressions”. In: Evolutionary Computation 29 (2 2021), pp. 211–237. DOI: 10.1162/evco_a_00278

    2021 doi

  69. [80]

    Genetic Programming in Python, with a scikit-learn inspired API: gplearn

    T. Stephens et al. “Genetic Programming in Python, with a scikit-learn inspired API: gplearn”. In: Docu- mentation at https://gplearn. readthedocs. io/en/stable/intro. html (2016)

    2016

  70. [81]

    GSR: A Generalized Symbolic Regression Approach

    T. Tohme, D. Liu, and K. Youcef-Toumi. “GSR: A Generalized Symbolic Regression Approach”. In:CoRR abs/2205.15569 (2022). DOI: 10.48550/ARXIV.2205.15569 . URL: https://doi.org/10. 48550/ARXIV.2205.15569

    2022 doi arXiv

  71. [82]

    A Unified Framework for Deep Symbolic Regression

    M. Landajuela, C. S. Lee, J. Yang, R. Glatt, C. P. Santiago, I. Aravena, T. N. Mundhenk, G. Mulc- ahy, and B. K. Petersen. “A Unified Framework for Deep Symbolic Regression”. In: Advances in Neu- ral Information Processing Systems 35 (NeurIPS 2022) . 2022. URL: http://papers.n...

    2022

  72. [83]

    Vertical Symbolic Regression via Deep Policy Gradient

    N. Jiang, M. Nasim, and Y . Xue. “Vertical Symbolic Regression via Deep Policy Gradient”. In: Proceed- ings of the 33rd International Joint Conference on Artificial Intelligence (IJCAI–24). 2024, pp. 5891–5899. DOI: 10.24963/ijcai.2024/651. URL: https://doi.org/10.24963/ijcai.2024/651

    2024 doi

  73. [84]

    A new formulation for symbolic regression to identify physico-chemical laws from experimental data

    P. Neumann, L. Cao, D. Russo, V . S. Vassiliadis, and A. A. Lapkin. “A new formulation for symbolic regression to identify physico-chemical laws from experimental data”. In: Chemical Engineering Journal 387 (2020), p. 123412. DOI: 10.1016/j.cej.2019.123412

    2020

  74. [85]

    Integration of neu- ral network-based symbolic regression in deep learning for scientific discovery

    S. Kim, P. Y . Lu, S. Mukherjee, M. Gilbert, L. Jing, V . ˇCeperi´c, and M. Solja ˇci´c. “Integration of neu- ral network-based symbolic regression in deep learning for scientific discovery”. In: (2019). DOI: 10 . 48550/arxiv.1912.04825

    2019 doi arXiv

  75. [86]

    Racing control variable genetic programming for symbolic regression

    N. Jiang and Y . Xue. “Racing control variable genetic programming for symbolic regression”. In:Proceed- ings of the AAAI Conference on Artificial Intelligence 38 (11 2024), pp. 12901–12909. DOI: 10.1609/ aaai.v38i11.29187

    2024

  76. [87]

    Multi-objective symbolic regression for physics-aware dynamic modeling

    J. Kubal ´ık, E. Derner, and R. Babu ˇska. “Multi-objective symbolic regression for physics-aware dynamic modeling”. In: Expert Systems With Applications 182 (2021), p. 115210. DOI: 10 . 1016 / j . eswa . 2021.115210

    2021

  77. [88]

    Symbolic Regression for Preci- sion LHC Physics

    M. Morales-Alvarado, D. Conde, J. Bendavid, V . Sanz, and M. Ubiali. “Symbolic Regression for Preci- sion LHC Physics”. In: arXiv 2412.07839 (2024). Uses PySR as the symbolic-regression engine. URL: https://arxiv.org/abs/2412.07839

    2024 arXiv

  78. [89]

    Data-driven discovery of tsallis-like distribution using symbolic regression in high-energy physics

    N. Makke and S. Chawla. “Data-driven discovery of tsallis-like distribution using symbolic regression in high-energy physics”. In: PNAS Nexus 3 (11 2024). DOI: 10.1093/pnasnexus/pgae467

    2024 doi

  79. [90]

    Deep symbolic regression for physics guided by units con- straints: toward the automated discovery of physical laws

    W. Tenachi, R. Ibata, and F. I. Diakogiannis. “Deep symbolic regression for physics guided by units con- straints: toward the automated discovery of physical laws”. In: The Astrophysical Journal 959 (2 2023), p. 99. DOI: 10.3847/1538-4357/ad014c

    2023 doi

  80. [91]

    Ai feynman: a physics-inspired method for symbolic regression

    S. Udrescu and M. Tegmark. “Ai feynman: a physics-inspired method for symbolic regression”. In:Science Advances 6 (16 2020). DOI: 10.1126/sciadv.aay2631. Draft: September 8, 2025 23

    2020 doi

  81. [92]

    Physics-informed learning of governing equations from scarce data

    Z. Chen, Y . Liu, and H. Sun. “Physics-informed learning of governing equations from scarce data”. In: Nature communications 12.1 (2021), p. 6136

    2021

  82. [93]

    Extrapolation and learning equations

    G. Martius and C. H. Lampert. “Extrapolation and learning equations”. In: arXiv preprint arXiv:1610.02995 (2016)

    2016 arXiv

  83. [94]

    Scalable Sparse Regression for Model Discovery: The Fast Lane to Insight

    M. Golden. “Scalable Sparse Regression for Model Discovery: The Fast Lane to Insight”. In: arXiv preprint arXiv:2405.09579 (2024)

    2024 arXiv

  84. [95]

    Large language models are zero- shot rankers for recommender systems

    Y . Hou, J. Zhang, Z. Lin, H. Lu, R. Xie, J. McAuley, and W. X. Zhao. “Large language models are zero- shot rankers for recommender systems”. In: European Conference on Information Retrieval . Springer. 2024, pp. 364–381

    2024

  85. [96]

    A survey on large language models for recommendation

    L. Wu et al. “A survey on large language models for recommendation”. In:arXiv preprint arXiv:2305.19860 (2023)

    2023 arXiv

  86. [97]

    Recommender systems in the era of large language models (llms)

    Z. Zhao et al. “Recommender systems in the era of large language models (llms)”. In: IEEE Transactions on Knowledge and Data Engineering (2024)

    2024

  87. [98]

    Decoding symbolism in language models

    M. Guo, R. Hwa, and A. Kovashka. “Decoding symbolism in language models”. In: Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) (2023). DOI: 10.18653/v1/2023.acl-long.186

    2023 doi

  88. [99]

    Open-source solution for evaluation and benchmarking of large language models for public health

    L. Espinosa, D. Kebaili, S. Consoli, K. Kalimeri, Y . Mejova, and M. Salath ´e. “Open-source solution for evaluation and benchmarking of large language models for public health”. In: (2025). DOI: 10.1101/ 2025.03.20.25324040

    2025

  89. [100]

    Applications of advanced natural language processing for clinical pharmacology

    J. C. Hsu et al. “Applications of advanced natural language processing for clinical pharmacology”. In: Clinical Pharmacology & Therapeutics 115 (4 2024), pp. 786–794. DOI: 10.1002/cpt.3161

    2024 doi

  90. [101]

    Large Language Models meet Collaborative Fil- tering: An Efficient All-round LLM-based Recommender System

    S. Kim, H. Kang, S. Choi, D. Kim, M. Yang, and C. Park. “Large Language Models meet Collaborative Fil- tering: An Efficient All-round LLM-based Recommender System”. In: arXiv preprint arXiv:2404.11343 (2024)

    2024 arXiv

  91. [102]

    Once: Boosting content-based recommendation with both open- and closed-source large language models

    Q. Liu, N. Chen, T. Sakai, and X.-M. Wu. “Once: Boosting content-based recommendation with both open- and closed-source large language models”. In: Proceedings of the 17th ACM International Conference on Web Search and Data Mining. 2024, pp. 452–461

    2024

  92. [103]

    Leveraging large language models for sequential recommendation

    J. Harte, W. Zorgdrager, P. Louridas, A. Katsifodimos, D. Jannach, and M. Fragkoulis. “Leveraging large language models for sequential recommendation”. In: Proceedings of the 17th ACM Conference on Rec- ommender Systems. 2023, pp. 1096–1102

    2023

  93. [104]

    Large Language Models for Next Point-of- Interest Recommendation

    P. Li, M. de Rijke, H. Xue, S. Ao, Y . Song, and F. D. Salim. “Large Language Models for Next Point-of- Interest Recommendation”. In: arXiv preprint arXiv:2404.17591 (2024)

    2024 arXiv

  94. [105]

    CoRAL: Collaborative Retrieval- Augmented Large Language Models Improve Long-tail Recommendation

    J. Wu, C.-C. Chang, T. Yu, Z. He, J. Wang, Y . Hou, and J. McAuley. “CoRAL: Collaborative Retrieval- Augmented Large Language Models Improve Long-tail Recommendation”. In:arXiv preprint arXiv:2403.06447 (2024)

    2024 arXiv

  95. [106]

    Large language models are competitive near cold-start recommenders for language-and item-based preferences

    S. Sanner, K. Balog, F. Radlinski, B. Wedin, and L. Dixon. “Large language models are competitive near cold-start recommenders for language-and item-based preferences”. In: Proceedings of the 17th ACM conference on recommender systems. 2023, pp. 890–896. Draft: September 8, 2025 24

    2023

  96. [107]

    ChatGPT for good? On opportunities and challenges of large language models for education

    E. Kasneci et al. “ChatGPT for good? On opportunities and challenges of large language models for education”. In: Learning and individual differences 103 (2023), p. 102274

    2023

  97. [108]

    Large language models in medicine

    A. J. Thirunavukarasu, D. S. J. Ting, K. Elangovan, L. Gutierrez, T. F. Tan, and D. S. W. Ting. “Large language models in medicine”. In: Nature medicine 29.8 (2023), pp. 1930–1940

    2023

  98. [109]

    A survey on large language models: Applications, challenges, limitations, and practical usage

    M. U. Hadi, R. Qureshi, A. Shah, M. Irfan, A. Zafar, M. B. Shaikh, N. Akhtar, J. Wu, S. Mirjalili, et al. “A survey on large language models: Applications, challenges, limitations, and practical usage”. In:Authorea Preprints (2023)

    2023

  99. [110]

    Using large language models in psychology

    D. Demszky et al. “Using large language models in psychology”. In: Nature Reviews Psychology 2.11 (2023), pp. 688–701

    2023

  100. [111]

    Why do pretrained language models help in downstream tasks? an analysis of head and prompt tuning

    C. Wei, S. M. Xie, and T. Ma. “Why do pretrained language models help in downstream tasks? an analysis of head and prompt tuning”. In:Advances in Neural Information Processing Systems34 (2021), pp. 16158– 16170

    2021

  101. [112]

    In-Context Symbolic Regression: Leverag- ing Large Language Models for Function Discovery

    M. Merler, K. Haitsiukevich, N. Dainese, and P. Marttinen. “In-Context Symbolic Regression: Leverag- ing Large Language Models for Function Discovery”. In: Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 4: Student Research Works...

    2024 doi

  102. [114]

    Sharlin and T

    S. Sharlin and T. R. Josephson. In Context Learning and Reasoning for Symbolic Regression with Large Language Models. 2025. arXiv: 2410.17448 [cs.CL]. URL: https://arxiv.org/abs/2410. 17448

    2025 arXiv

  103. [115]

    Y . Li, W. Li, L. Yu, M. Wu, J. Liu, W. Li, S. Wei, and Y . Deng. MLLM-SR: Conversational Symbolic Regression base Multi-Modal Large Language Models . 2024. arXiv: 2406 . 05410 [cs.AI]. URL: https://arxiv.org/abs/2406.05410

    2024 arXiv

  104. [116]

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

    1992

  105. [117]

    Distilling free-form natural laws from experimental data

    M. Schmidt and H. Lipson. “Distilling free-form natural laws from experimental data”. In: Science 324.5923 (2009), pp. 81–85. DOI: 10.1126/science.1165893

    2009 doi

  106. [118]

    Goldstein, C

    H. Goldstein, C. P. Poole, and J. L. Safko. Classical Mechanics. 3rd ed. Addison-Wesley, 2001. ISBN : 9780201657029

    2001

  107. [119]

    J. D. Jackson. Classical Electrodynamics. 3rd ed. Wiley, 1999. ISBN : 9780471309321

    1999

  108. [120]

    Improving symbolic regression with interval arithmetic and linear scaling

    M. Keijzer. “Improving symbolic regression with interval arithmetic and linear scaling”. In: European Conference on Genetic Programming. V ol. 2610. LNCS. 2003, pp. 70–82

    2003

  109. [121]

    Semantically-based crossover in ge- netic programming: application to real-valued symbolic regression

    N. Q. Uy, N. X. Hoai, M. O’Neill, R. McKay, and E. Galv ´an-L´opez. “Semantically-based crossover in ge- netic programming: application to real-valued symbolic regression”. In:Genetic Programming and Evolv- able Machines 12.2 (2011), pp. 91–119. Draft: September 8, 2025 25

    2011

  110. [122]

    Order of nonlinearity as a complexity measure for models generated by symbolic regression via Pareto genetic programming

    E. J. Vladislavleva, G. F. Smits, and D. den Hertog. “Order of nonlinearity as a complexity measure for models generated by symbolic regression via Pareto genetic programming”. In: IEEE Transactions on Evolutionary Computation 13.2 (2009), pp. 333–349

    2009

  111. [123]

    A Robust Symbolic Regression Benchmark with Noise, Extrapolation, and Overfitting Controls

    P. A. Reinbold, D. Beloin, and L. Haghverdi. “A Robust Symbolic Regression Benchmark with Noise, Extrapolation, and Overfitting Controls”. In: Genetic Programming and Evolvable Machines 22 (2021), pp. 205–239. DOI: 10.1007/s10710-020-09384-2

    2021 doi

  112. [124]

    LLM-SR: Guiding Symbolic Regression with Language Models

    N. Shojaee and et al. “LLM-SR: Guiding Symbolic Regression with Language Models”. In: arXiv preprint arXiv:2502.XXXXX (2025)

    2025

  113. [125]

    FunSearch: Bridging Program Synthesis and Discovery with LLMs

    B. Romera-Paredes and et al. “FunSearch: Bridging Program Synthesis and Discovery with LLMs”. In: Nature (2023)

    2023

  114. [126]

    In-Context Symbolic Regression with LLMs

    M. Merler and et al. “In-Context Symbolic Regression with LLMs”. In: arXiv preprint arXiv:2401.XXXXX (2024)

    2024

  115. [127]

    Scaling Language Models: Methods, Analysis & Interpretation

    J. Rae and et al. “Scaling Language Models: Methods, Analysis & Interpretation”. In: arXiv preprint arXiv:2109.04486 (2021)

    2021 arXiv

  116. [128]

    GPT-4 Technical Report

    OpenAI. “GPT-4 Technical Report”. In: arXiv preprint arXiv:2303.08774 (2023)

    2023 arXiv

  117. [129]

    FrugalGPT: How to Use a Large Language Model Cheaply and Efficiently

    Y . Chen and et al. “FrugalGPT: How to Use a Large Language Model Cheaply and Efficiently”. In: arXiv preprint arXiv:2305.05176 (2023)

    2023 arXiv

  118. [130]

    Early Stopping Against Label Noise Without Validation Data

    S. Yuan, L. Feng, and T. Liu. “Early Stopping Against Label Noise Without Validation Data”. In: Proceed- ings of the 12th International Conference on Learning Representations (ICLR). 2024

    2024

  119. [131]

    PromptBench: Towards Evaluating the Robustness of Large Language Models on Adversar- ial Prompts

    K. Zhu et al. “PromptBench: Towards Evaluating the Robustness of Large Language Models on Adversar- ial Prompts”. In: arXiv preprint arXiv:2306.04528 (2023)

    2023 arXiv

  120. [132]

    On the Worst Prompt Performance of Large Language Models

    B. Cao, D. Cai, Z. Zhang, Y . Zou, and W. Lam. “On the Worst Prompt Performance of Large Language Models”. In: Advances in Neural Information Processing Systems (NeurIPS 2024). 2024

    2024

  121. [133]

    Quantifying Language Models’ Sensitivity to Spurious Features in Prompt Design, or: How I Learned to Start Worrying about Prompt Formatting

    M. Sclar, Y . Choi, Y . Tsvetkov, and A. Suhr. “Quantifying Language Models’ Sensitivity to Spurious Features in Prompt Design, or: How I Learned to Start Worrying about Prompt Formatting”. In: arXiv preprint arXiv:2310.11324 (2023)

    2023 arXiv

  122. [134]

    Semantic Consistency for Assuring Reliability of Large Language Models

    H. Raj, V . Gupta, D. Rosati, and S. Majumdar. “Semantic Consistency for Assuring Reliability of Large Language Models”. In: arXiv preprint arXiv:2308.09138 (2023)

    2023 arXiv

  123. [135]

    ProSA: Assessing and Understanding the Prompt Sensitivity of LLMs

    J. Zhuo, S. Zhang, X. Fang, H. Duan, D. Lin, and K. Chen. “ProSA: Assessing and Understanding the Prompt Sensitivity of LLMs”. In: Findings of the Association for Computational Linguistics: EMNLP

  124. [136]

    A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge

    L. S. Keren, A. Liberzon, and T. Lazebnik. “A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge”. In: Scientific Reports 13.1 (2023), p. 1249

    2023

  125. [137]

    Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery

    Y . Matsubara, N. Chiba, R. Igarashi, and Y . Ushiku. “Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery”. In: Journal of Data-centric Machine Learning Research (2024). URL: https://openreview.net/forum?id=qrUdrXsiXX. Draft: September 8, 2025 26

    2024

  126. [138]

    The Metric is the Message: Benchmark- ing Challenges for Neural Symbolic Regression

    A. Bertschinger, Q. T. Davis, J. P. Bagrow, and J. C. Bongard. “The Metric is the Message: Benchmark- ing Challenges for Neural Symbolic Regression”. In: Machine Learning and Knowledge Discovery in Databases: Research Track (ECML PKDD 2023, Part IV) . V ol. 14172. Lecture Note...

    2023 doi

  127. [139]

    Benchmarking symbolic regression constant optimization schemes

    L. G. A. dos Reis, V . L. P. S. Caminha, and T. J. P. Penna. “Benchmarking symbolic regression constant optimization schemes”. In: arXiv preprint arXiv:2412.02126 (2024)

    2024 arXiv

  128. [140]

    R. Ma, J. Luijkx, Z. Ajanovic, and J. Kober. ExploRLLM: Guiding Exploration in Reinforcement Learning with Large Language Models. 2024. arXiv: 2403.09583 [cs.RO] . URL: https://arxiv.org/ abs/2403.09583

    2024 arXiv

  129. [141]

    Z. Zhou, B. Hu, C. Zhao, P. Zhang, and B. Liu. Large Language Model as a Policy Teacher for Training Reinforcement Learning Agents. 2024. arXiv: 2311.13373 [cs.AI]. URL: https://arxiv.org/ abs/2311.13373

    2024 arXiv

  130. [142]

    Sensitivity analysis in optimization and reliability problems

    E. Castillo, R. M ´ınguez, and C. Castillo. “Sensitivity analysis in optimization and reliability problems”. In: Reliability Engineering & System Safety 93.12 (2008), pp. 1788–1800

    2008

  131. [143]

    Pre-train Prompt Tuning: Towards Effective LLM Guidance

    X. Liu, Y . Zhang, and J. Wang. “Pre-train Prompt Tuning: Towards Effective LLM Guidance”. In: arXiv preprint arXiv:2302.00000 (2023)

    2023

  132. [144]

    Exploiting Cloze Questions for Few-Shot Text Classification and Natural Language Inference

    T. Schick and H. Sch ¨utze. “Exploiting Cloze Questions for Few-Shot Text Classification and Natural Language Inference”. In: Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics (EACL). 2021, pp. 255–269

    2021

  133. [145]

    Self-Consistency Improves Chain of Thought Reasoning in Language Models

    X. Wang, J. Wei, D. Schuurmans, Q. Le, E. Chi, S. Narang, A. Chowdhery, D. Zhou, et al. “Self-Consistency Improves Chain of Thought Reasoning in Language Models”. In:arXiv preprint arXiv:2203.11171(2022)

    2022 arXiv

  134. [146]

    Benchmarking state-of-the-art symbolic regression algorithms

    J. Zegklitz and P. Po ˇs´ık. “Benchmarking state-of-the-art symbolic regression algorithms”. In: Genetic Programming and Evolvable Machines 22.1 (2021), pp. 5–33

    2021

  135. [147]

    Contemporary symbolic regression methods and their relative performance

    W. La Cava, V . Moosavi, R. Taguchi, and J. H. Moore. “Contemporary symbolic regression methods and their relative performance”. In: Genetic Programming and Evolvable Machines22.4 (2021), pp. 385–409. DOI: 10.1007/s10710-021-09400-4

    2021 doi

  136. [148]

    Causal benchmarks for physical reasoning

    S. G. Srivastava et al. “Causal benchmarks for physical reasoning”. In: NeurIPS Datasets and Benchmarks Track (2022). URL: https://openreview.net/pdf?id=eIm4q5yNzzl

    2022

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.