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REVIEW 3 major objections 5 minor 4 cited by

DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read LLM symbolic regression is improved by a closed loop of data insight and reflective idea extraction.

desk verdict A plausible incremental extension of LLM-SR with a serious internal inconsistency in the headline Oscillator 1 result. read the letter →

arxiv 2506.04282 v1 pith:FL6E2B4S submitted 2025-06-04 cs.LG

Runxiang Wang , Boxiao Wang , Kai Li , Yifan Zhang , Jian Cheng This is my paper
classification cs.LG
keywords symbolicregressionlargelanguagemodelsscientificequationdiscoverydata-awareinsightinductiveideaextractionclosed-loopreasoningskeletongenerationbenchmarkevaluation
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

This paper argues that large language models can discover scientific equations much more reliably when their generation is guided by two additional reasoning streams: explicit analysis of the raw data, and distilled lessons from previous successes and failures. It introduces DrSR, a framework in which a data-analysis module inspects a sample of input-output pairs and residual patterns, an idea-extraction module categorizes each candidate equation as positive, negative, or invalid and summarizes reusable heuristics, and both streams feed back into the equation generator. On six benchmarks spanning physics, chemistry, biology, and materials science, DrSR reports higher valid equation rates, lower normalized mean squared error, better out-of-distribution generalization, and faster convergence than LLM-SR, PySR, and other baselines. A sympathetic reading of the result is that the bottleneck in LLM-based symbolic regression is not generation ability alone but the absence of structured observation and self-correction, and that closing this loop lets the model explore the expression space far more efficiently.

What carries the argument

The load-bearing mechanism is a closed loop of three LLM-instantiations sharing one backbone: $\pi_{\text{data}}$ (data-aware insight), $\pi_{\text{idea}}$ (inductive idea extraction), and $\pi_{\text{main}}$ (equation skeleton generation). $\pi_{\text{data}}$ uniformly samples 100 points from the dataset, adds residuals $r_{t,i} = y_i - f^*(x_i)$ from the best current equation, and produces structured descriptions of monotonicity, nonlinearity, correlation, and inter-variable relations; these insights are refreshed every time a better fit appears. $\pi_{\text{idea}}$ labels each candidate as positive, negative, or invalid, extracts a one-line lesson (e.g., avoid array broadcast errors, include interaction terms), and stores it in a JSON idea library sampled from the most recent half. Both $\mathcal{D}$ and $\mathcal{I}$ are injected into the prompt of $\pi_{\text{main}}$, which generates skeletons optimized by BFGS; the loop repeats for up to 1000 iterations. The paper's Bayesian framing makes the work of this machinery precise: it converts the static prior $p_{\text{LLM}}(f)$ into a dynamic, context-aware prior $p_{\text{LLM}}(f \mid \mathcal{D}, \mathcal{I})$ that sharpens as the search proceeds.

What would settle it

Run DrSR with the data-insight output replaced by random or deliberately wrong structural hints (e.g., 'possible exponential relationship' when truth is polynomial) across the same six benchmarks; if accuracy and valid rate do not drop substantially, the insight module is not the driver of the reported gains. Alternatively, run the search on a dataset whose 100-sample subset is unrepresentative (e.g., points concentrated in one region) and check whether DrSR fails to recover the true equation while a full-data variant succeeds.

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Extended reading notes

Core claim

The central claim is that coupling data-aware insight with inductive idea extraction transforms LLM-based symbolic regression from a prior-driven generator into a self-correcting scientific process, and that this yields state-of-the-art accuracy on the tested tasks. Concretely, DrSR's data module samples 100 points (augmented with residuals of the current best equation) and emits structured descriptions such as 'possible cubic relationship in the range -0.2 to 0.2'; the idea module classifies each generated skeleton as positive, negative, or invalid and stores concise lessons in a JSON library; the main generator receives both insight and ideas in its prompt. The paper reports that DrSR achieves an NMSE of $1.80\times10^{-12}$ on Oscillator 2 versus $4.59\times10^{-5}$ for LLM-SR, exceeds 90% accuracy under strict tolerances on several tasks, maintains higher valid equation rates (0.4–0.6 vs sub-0.3 on complex datasets), and converges in 1000 iterations where baselines need 2000. The authors interpret this through a Bayesian reformulation in which the generative prior becomes $p_{\text{LLM}}(f \mid \mathcal{D}, \mathcal{I})$, conditioned on evolving data insight and idea library, rather than a static language-based prior.

Load-bearing premise

The data-aware insight module, using only 100 sampled points plus residuals from the current best equation, produces structural hints (e.g., 'possible cubic relationship') that are accurate enough to steer the skeleton search toward the true expression; if the LLM's data analysis is noisy or misleading, the reflective loop could guide the search away from the correct region, and the method's advantage would vanish or reverse.

Editorial extensions

If this is right

  • DrSR raises the valid equation rate, cutting the fraction of generations wasted on syntax errors, numerical overflows, or variable mismatches, and thereby improves sample efficiency under a limited iteration budget.
  • The reported accuracy and NMSE gains—including near-exact recovery on Oscillator 2 and 90%+ tolerance accuracy on several tasks—would make DrSR a practical tool for extracting interpretable laws from experimental data in materials, biology, and chemistry.
  • Strong out-of-distribution performance (e.g., OOD NMSE $2.80\times10^{-11}$ on Oscillator 2 where uDSR degrades to $9.59\times10^{3}$) suggests the discovered expressions capture latent structure rather than memorizing the training distribution.
  • Because the loop updates insight from residuals, the framework naturally refines its structural understanding from coarse global patterns to finer local and higher-order structure over iterations, mirroring a scientist's deepening understanding.
  • The Bayesian reinterpretation implies the method can be seen as adaptive prior learning; this could transfer to other LLM-guided search tasks where the prior is initially weak and data is plentiful.

Reading between the lines

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

  • The 100-point sampling plus residual refinement suggests a general recipe: a small, actively-updated data summary may be sufficient to steer an LLM through a combinatorial search space; one testable extension is to vary sample size and noise to find when the insight signal breaks.
  • The idea library mechanism is essentially a structured memory of failure modes; this pattern could generalize to other generative settings (e.g., code synthesis, PDE discovery) where reflection on invalid outputs is currently ignored.
  • The paper's framing implies that the value of an LLM in scientific discovery is not just its internal knowledge but its ability to integrate external observations; one could probe this by measuring how much performance degrades when the data insight is computed by a different, weaker model.
  • Since the ablation shows both components contribute and their removal degrades performance, a natural next step left implicit is to study how the insights and ideas themselves evolve—whether they converge to the true structure or merely to a local optimum of the prompt distribution.
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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

3 major / 5 minor

Summary. The paper proposes DrSR, an LLM-based symbolic regression framework that extends LLM-SR with two reasoning modules: a data-aware insight module (πdata) that summarizes structural patterns from sampled data and residual-augmented data, and an inductive idea extraction module (πidea) that distills positive, negative, and error experiences into an evolving idea library. The main generation loop prompts an LLM to propose equation skeletons, fits their parameters with BFGS, and feeds back both insights and ideas to improve subsequent proposals. Experiments are reported on six tasks (two nonlinear oscillators, E. coli growth, stress-strain, two LSR-Transform tasks, and one LSR-Synth task) with Mixtral-8x7B and LLaMA3.1 backbones, against gplearn, PySR, DSR, uDSR, LaSR, and LLM-SR. The paper claims consistently higher accuracy (ACC_tau), lower NMSE, faster convergence, and higher valid solution rates, and it includes a Bayesian reinterpretation of the search objective.

Significance. The proposed architecture is a natural and plausible improvement over LLM-SR: the two modules are clearly motivated, the ablation on Oscillator 2 shows that each contributes, and the valid-rate improvement is practically important for LLM-based search. If the reported numbers are reliable, DrSR would be a strong state-of-the-art LLM-based symbolic regression method. The paper's strengths include a clean algorithmic description (Algorithms 1-3), comparisons against six diverse baselines, and a useful discussion of the insight/idea loops with prompt examples. Its main weaknesses are statistical: no repeated seeds or error bars anywhere, a small and partially selected benchmark set, and one internal contradiction in the headline Oscillator 1 result that prevents verification of the reported NMSE. No code or data release is mentioned, which further weakens reproducibility.

major comments (3)
  1. [Appendix D / Table 1] The best-equation display for Oscillator 1 is inconsistent with the reported NMSE. Appendix D states the ground truth as dv/dt = 0.8 sin(x) - 0.5 x v - 0.5 v^3 - 0.2 x^3 - x cos(x), and shows the best DrSR equation as the same expression without the -x cos(x) term. Yet Table 1 reports DrSR (Mixtral) NMSE = 3.14e-7 and DrSR (Llama) NMSE = 5.40e-7 on this task. With the stated initial conditions x0 = 0.5, v0 = 0.5 over t in [0,50], omitting -x cos(x) leaves residuals whose magnitude is not negligible relative to the target variance; an NMSE near 1e-7 would require the omitted term to be essentially zero throughout the test set, which is implausible for this simulation. The authors must reconcile the displayed equation with the reported metric: either the equation is incomplete, the data generation differs from Appendix B.1, or the metric was computed against a different target. As written, the headline accuracy numbers for Oscillator 1, and by extension the evaluation pipeline for the other five tasks, cannot be verified.
  2. [Section 4.4 / Table 1 / Figures 2-4] Every empirical result is reported as a single point estimate with no repeated seeds, confidence intervals, or error bars. The LLM backbones are stochastic and Appendix E acknowledges that outputs can be verbose and variable; BFGS can also converge to different local optima from different skeletons. Under these sources of variance, margins such as DrSR (Llama) Acc = 3.64% vs LLM-SR (Llama) Acc = 1.36% on E. coli growth, or the valid-rate differences in Figure 4, may be within run-to-run noise. Since the central claim is empirical superiority, the paper should provide distributions over at least 3-5 independent runs per configuration for Table 1 and for the ablation in Section 5.5.
  3. [Section 5.2 / Figure 2] The generalization claim is not assessable because the paper never defines the out-of-distribution (OOD) split. Figure 2 plots ID vs OOD NMSE, but the text does not say how OOD inputs are constructed (e.g., extrapolation in time, new parameter regimes, or extended variable ranges). Without an explicit definition, the claim that DrSR 'learns latent rules behind data rather than overfitting' cannot be checked, and the dramatic OOD improvements (e.g., Oscillator 2 OOD NMSE 2.80e-11 vs uDSR 9.59e3) cannot be reproduced. Please add a precise description of the OOD protocol for each dataset.
minor comments (5)
  1. [Table 1] The header 'GPlern' is a typo for 'gplearn'; also the column 'LSR-Transform-2Avg' should be named consistently with the dataset description in Section 4.1 ('LSR-Transform').
  2. [Section B.5] The reaction-kinetics equation contains '0.1899_z' with an unexplained subscript z; this appears to be a typographical or formatting artifact and should be fixed.
  3. [Section 3.4] The Bayesian formulation in Equations (1)-(2) is purely interpretive and no quantitative priors or likelihoods are specified; adding a sentence clarifying that this is a high-level reinterpretation rather than a computational model would avoid overstating its role.
  4. [Appendix A / reproducibility] No code or data availability statement is provided. Given the dependence on specific prompts (Appendix C), dataset splits, and LLM decoding parameters, releasing the implementation and data-generation scripts is necessary for reproducibility.
  5. [Appendix D / Figure 11] The text refers to terms 'highlighted in pink', but the figure in the submitted PDF appears without color highlighting; please ensure the final version preserves the highlighting or describes the marked terms in text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: DrSR's stated gains rest on held-out test evaluation, not on a fitted input renamed as a prediction or on a self-citation chain.

full rationale

DrSR is an empirical LLM-based symbolic-regression pipeline; it does not contain a mathematical derivation that reduces to its own inputs. The BFGS-fitted coefficients are optimized on training data and then evaluated on ID/OOD test metrics described in Sections 4.3 and Appendix F, so the reported NMSE/ACC values are not definitionally equal to the fitting objective. The data-aware insight module (Section 3.1, Algorithm 3) consumes training samples and residuals, which is ordinary supervised fitting rather than a prediction forced by construction; the residual loop updates the skeleton search, but this is a heuristic optimization loop, not a tautology. The Bayesian reformulation in Section 3.4 is explicitly non-objective-changing ('This formulation does not change the task objective'), so it cannot be a load-bearing circular step. The paper's citations to LLM-SR and LLM-SRBench are to other author groups and are used for benchmark construction, not to import a contested uniqueness claim or an unverified ansatz. No fitted parameter is renamed as a prediction, and no result is justified solely by a self-citation. One in-scope inconsistency exists outside circularity: Appendix D reports the best DrSR equation for Oscillator 1 as omitting the ground-truth '-x cos(x)' term while Table 1 lists NMSE 3.14e-7, which is a reproducibility and accuracy-reporting concern, but it does not constitute a reduction of the claim to its inputs. Because the central comparisons are against external baselines on external benchmark tasks and the reported test metrics are not constructed to equal the training fit, the derivation is self-contained with respect to circularity.

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

The method introduces no new physical entities. The free parameters are hyperparameters chosen by hand; the axioms are the correctness of the benchmark labels, the LLM's prompt-following ability, and the adequacy of BFGS for parameter fitting.

free parameters (4)
  • data sample size for insight = 100
    Number of input-output pairs sampled for data analysis; chosen by hand, affects the quality of structural hints.
  • candidate skeletons per iteration = 4
    Number of equation skeletons sampled per iteration; larger values increase compute, smaller values may reduce coverage.
  • proportion of recent ideas sampled (lambda) = 0.5
    Fraction of most recent idea entries per category used in prompts; controls recency versus diversity.
  • insight sampling temperature and top-k/top-p = temperature 0.6, top-k 30, top-p 0.3
    Decoding parameters for the data-analysis LLM; chosen ad hoc without sensitivity analysis.
assumptions (3)
  • domain assumption The benchmark ground-truth expressions are correct and representative of the scientific domains.
    Used to define ACC_tau and NMSE targets; if labels are wrong, the reported comparisons lose meaning.
  • ad hoc to paper The LLM backbones can follow the structured prompts and generate useful data insights and ideas.
    The entire method depends on this capability; Section 3.1 and Appendix C assume the model produces accurate structural descriptions rather than hallucinations.
  • ad hoc to paper BFGS converges to adequate parameter values for most skeletons.
    Parameter optimization is done with BFGS (Section 4.4); if it fails, candidate equations are discarded as invalid or scored poorly.

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Cite this review

Pith. "Pith review of DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience." pith.science (2026). https://pith.science/paper/FL6E2B4S

@misc{pith2026250604282,
  author       = {Pith},
  title        = {Pith review of: DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FL6E2B4S}},
  note         = {Machine review of arXiv:2506.04282}
}
read the original abstract

Symbolic regression is a fundamental tool for discovering interpretable mathematical expressions from data, with broad applications across scientific and engineering domains. Recently, large language models (LLMs) have demonstrated strong performance in this task, leveraging embedded scientific priors and reasoning capabilities to surpass traditional methods. However, existing LLM-based approaches, such as LLM-SR, often over-rely on internal priors, lacking explicit data understanding and systematic reflection during equation generation. To address these limitations, we propose DrSR (Dual Reasoning Symbolic Regression), a framework that combines data-driven insight with reflective learning to enhance both robustness and discovery capability. Specifically, DrSR guides LLMs to analyze structural relationships (e.g., monotonicity, nonlinearity, and correlation) within the data to generate structured descriptions. Simultaneously, it monitors equation performance and establishes a feedback loop to refine subsequent generations. By integrating data understanding and generation reflection in a closed loop, DrSR enables more efficient exploration of the symbolic expression space. Experiments across interdisciplinary datasets in physics, chemistry, biology, and materials science demonstrate that DrSR substantially improves the valid equation rate and consistently outperforms both classical and recent LLM-based methods in terms of accuracy, generalization, and search efficiency. These results underscore its potential for scientific equation discovery.

Figures

Figures reproduced from arXiv: 2506.04282 by the authors.

Figure 1
Figure 1. Overview of the DrSR framework. (a) Data-aware Insight: the LLM analyzes variable relationships to derive structured, data-informed insights; (b) Inductive Idea Extraction: based on equation fitness, the LLM summarizes successful and failed patterns into an evolving idea library; (c) Equation Generation and Selection: guided by (a) and (b), the LLM generates new equation skeletons, optimizes parameters, and caches t… view at source ↗
Figure 2
Figure 2. Generalization across scientific domains under ID and OOD settings. DrSR achieves the lowest NMSE across all tasks and settings, indicating strong generalization performance. We further assess generalization by evaluating performance under both in-distribution (ID) and out￾of-distribution (OOD) settings [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Training convergence comparison. DrSR demonstrates faster convergence and lower final error across all tasks, consistently outperforming baselines even in early-stage iterations. 5.4 Valid Solution Rate Analysis To further explain the superior efficiency of DrSR, we evaluate the valid solution rate—the proportion of generated equations that are syntactically valid, compilable, and executable [PITH_FULL_IMAGE:figure… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Valid solution rate comparison. We report the proportion of syntactically valid, compilable, and evaluable equations produced by DrSR and LLM-SR. 5.5 Ablation Study We conduct ablation experiments on the Oscillator 2 task using Mixtral backbone to evaluate the contribu…
Figure 5
Figure 5. Figure 5: Ablation results on the Oscillation 2 problem, showing the impact of Data-aware Insight and Inductive Idea Extraction on the performance of DrSR. Impact of Data-aware Insight. We vary the probability p ∈ {0, 0.5, 1.0} with which the main LLM references structured insig…
Figure 6
Figure 6. Figure 6: Prompt for Data-aware Insight. The optimized function skeleton you just answered scored higher. Please summarize useful experience. STRICTLY follow these rules: 1. Use the exact phrasing "when seeking for the mathematical function skeleton that represents {dependent_na…
Figure 7
Figure 7. Figure 7: Prompt for Inductive Idea Extraction. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Initial Data-aware Insight. "output_format": { "analysis": { "independent_to_dependent_relationships": { "x ": [ "The relationship between x and a is a non-linear one, with a possible cubic relationship in the range of -0.2 to 0.2. However, the relationship becomes mor…
Figure 9
Figure 9. Figure 9: Final Data-aware Insight. The following are previous experience summaries that can help you provide better answers. experience 1： When I optimize the equation framework, in order to avoid the output failure of the generated function framework, I need to ensure that the…
Figure 10
Figure 10. Figure 10: Final Inductive Idea Extraction. D: Results Presentation [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Performance trajectory of DrSR with the best-performing equation programs discovered [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

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    when seeking for the mathematical function skeleton that represents {dependent_name_in_prompt} in {problem_name_in_prompt}, I can

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    You need to make your answer as concise as possible Prompt 4 - Inductive Idea Extraction (Positive case) Prompt 5 - Inductive Idea Extraction (Negative case) Prompt 6 - Inductive Idea Extraction (Invalid case) Figure 7: Prompt for Inductive Idea Extraction. 17 ### The followin...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.