Beyond Equation Search: Conjecture-Guided Discovery with SCOPE and ClosureBench

Yuxiao Hu; Qian Li; Ding Wang; Siheng Chen; Linfeng Zhang; Jinyue Yan; Yuntian Chen

cs.AIOct 13, 2026

Beyond Equation Search: Conjecture-Guided Discovery with SCOPE and ClosureBench

Authors: Yuxiao Hu, Qian Li, Ding Wang, Siheng Chen, Linfeng Zhang, Jinyue Yan, Yuntian Chen

Organizations: The Hong Kong Polytechnic University, Hong Kong, China · Eastern Institute of Technology, Ningbo, China · Shanghai Jiao Tong University, Shanghai, China · DP Technology, Beijing, China

Abstract

Most symbolic regression methods optimize candidate equations, but improving an equation's fit does not establish whether its underlying scientific conjecture should be retained, revised, or rejected. A conjecture may admit multiple mathematical realizations, and similarly accurate equations may imply different explanations. This motivates a shift from refining individual expressions to developing and testing the conjectures that guide them. We introduce SCOPE, a framework that sustains multiple conjecture-driven research trajectories. Multiple scientist agents independently pursue conjectures through equation modeling, question formulation, and active experimentation, then exchange evidence to revise their conjectures and research directions. Their findings are consolidated into a scientific report linking models, conclusions, and evidence. Existing equation-discovery benchmarks primarily assess recovered expressions, leaving the evidence and scientific conclusions established through the investigation underexamined. We therefore introduce ClosureBench, an interactive benchmark of 60 closure-discovery tasks across five scientific domains. ClosureBench focuses on closure-discovery tasks to give equation discovery a concrete scientific purpose. Each task embeds an unknown algebraic relation in a known PDE system, where agents control experimental conditions in OpenFOAM to acquire new data for equation discovery. Beyond equation accuracy, ClosureBench proposes three metrics to evaluate the discovery process: evidence faithfulness, mechanism insight, and experimental effectiveness. Compared with diverse baselines, SCOPE achieves higher equation accuracy on both ClosureBench and a real-world scientific discovery case, while producing more accurate, reliable, and effective scientific discoveries.

Figures & tables

Figure 1: Overview of SCOPE. SCOPE organizes scientific discovery around parallel, conjecture-driven investigations. Scientists use equation modeling and active experiments to test and revise conjectures, while roundtable discussions exchange evidence and guide further exploration. The final report connects quantitative models with supported scientific findings and open questions.

Method / Framework	Primary object of refinement	Symbolic equation modeling	Active experimentation	Parallel research streams	Primary output
Equation search ( Cranmer, 2023 ; Shojaee et al., 2025a )	Equations	$\checkmark$	—	—	Equations
Agentic equation discovery ( Xia et al., 2025 )	Equations	$\checkmark$	—	—	Equations
Active structure discovery ( Kabra et al., 2026 ; Abhyankar et al., 2026 )	Equations / network structures	$\checkmark$	$\checkmark$	—	Equations / network structures
Scientific research agents ( Gottweis et al., 2026 ; Mitchener et al., 2025 )	Hypotheses / research plans	—	—	—	Research proposals
SCOPE (Ours)	Conjecture-driven research trajectories	$\checkmark$	$\checkmark$	$\checkmark$	Equations + scientific report

Table 1: Comparison of scientific discovery frameworks across key design dimensions.

Panel of Figure 2 — Figure 2: ID Acc@1% with Qwen3.6-35B-A3B at 0, 6, 12, 18, 24, and 30 additional simulations. The six-call spacing follows SCOPE’s two-round roundtables in these runs (three scientists, one simulation each per round). Left: ReAct, Reflexion-Exp, and SCOPE under the same simulation-call budget. Right: SCOPE and the budget-matched Single-Conjecture ablation.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	$\|\mathcal{F}_{i}^{r}\|$	$\|\widetilde{\mathcal{F}}_{i}^{r}\|$	$K$	ID Acc@1% $\uparrow$	OOD Acc@1% $\uparrow$	MI $\uparrow$
Prior-Narrow	2	6	4	38.89	22.22	0.34
Default	4	12	4	41.67	25.00	0.38
Prior-Wide	8	24	4	40.56	25.56	0.36
Retention-Narrow	4	12	2	39.44	21.67	0.31
Retention-Wide	4	12	8	41.11	23.89	0.35

Appendix

Table 3: Sensitivity of SCOPE to candidate-set capacities with

M=3

Scientific domain	Representative task	Closure output	Observable closure inputs	Role in the PDE	Experimental controls
Reactive transport	One-sided heated reactive channel	$S$	$C,T$	Reaction source in the concentration balance	Inlet concentration, wall temperature, initial concentration
Energy and thermal sciences	Heterogeneous electrothermal heating	$Q_{J}$	$T,E,\chi$	Volumetric Joule source in the heat equation	Applied voltage, thermal level, composition-gradient amplitude
Environmental mass transfer	Counter-current spray-tower absorption	$S_{\mathrm{abs}}$	$C_{g},C_{b},d,T$	Coupled sinks in gas-solute and liquid-reagent balances	Inlet solute loading, nozzle diameter, inlet gas temperature
Materials processing and rheology	Flow-induced crystallization	$R_{\psi}$	$\psi,T,g$	Crystallinity source and latent-heat source	Inlet velocity, crystallinity seed, wall temperature
Particulate and porous-media flow	Mobile–immobile dual-porosity exchange	$g$	$C_{m},C_{i},r,\epsilon$	Opposite exchange source and sink in two pore regions	Darcy flux, inlet mobile concentration, initial immobile concentration, mean porosity

Appendix

Table 4: Representative ClosureBench tasks from five scientific domains. Experimental controls are simulator settings available to the investigating method; the closure inputs are the local variables used by the symbolic expression.

Evaluation task	Items	Human–human	LLM–human
EF evidence support	200	94.5%	92.0%
MI fact matching and recovery	150	92.7%	88.0%

Appendix

Table 5: Agreement between the LLM evaluator and human annotations.

Figure 3: Complete prompt used by the method-blind Evidence Faithfulness evaluator. A full-width boxed rendering of the Evidence Faithfulness evaluator prompt, including its system instruction, strict JSON output schema, report-claim input, and cited trajectory-evidence input.

Figure 4: Complete prompt used by the method-blind Mechanism Insight evaluator. A full-width boxed rendering of the Mechanism Insight evaluator prompt, including its three-level claim–fact recovery rubric and strict JSON output schema.

ID	Paper-anchored mechanism fact	Judgment	Location in the official JFM PDF
F01	Mirror symmetry in the signed cross-wake coordinate.	Complete	§2.3.2, (2.15), PDF p. 10
F02	Two comparable off-axis maxima in the transverse force profile.	Complete	§2.3.2, (2.15), fig. 6(b), PDF pp. 10–11
F03	A centerline response below the off-axis maxima, producing a central trough.	Complete	§2.3.2, fig. 6(b), PDF p. 11
F04	An approximately constant off-axis peak location within the localized forcing region.	Complete	Fig. 7(d), §3.1, PDF pp. 12–13
F05	An approximately constant effective transverse lobe width.	Partial	Fig. 7(d), §3.1, PDF pp. 12–13
F06	Monotonic downstream weakening of force amplitude through the active interval.	Complete	Fig. 7(c), §3.1, PDF pp. 12–13

Appendix

Table 6: Paper-anchored mechanism facts used to evaluate the JFM scientific report.

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ERBench: A Benchmark and Testsuite for Equation Discovery Algorithms

Paul Kahlmeyer, Henrik Voigt, Michael Habeck, Joachim Giesen

Department of Computer Science University of Jena Inselplatz 5, 07743 Jena, Germany

Benchmark Design Scientific Discovery Benchmark Validity Symbolic Regression

Open paper AskPDF arXiv

Equation discovery aims to automate the discovery of scientific models in the form of mathematical equations from data. Technically, equation discovery is implemented by symbolic regression algorithms. Performance of symbolic regression for equation discovery is measured along two dimensions: Prediction accuracy on test data, and recovery of known groundtruth formulas. For standard regression, accuracy is typically measured on in-domain test data, for instance, by splitting a data set randomly into training and test data. While this makes sense for in-domain interpolation, which is the common goal in ordinary regression, it can be a misleading proxy for true model discovery and generalization. The obvious alternative is to measure out-of-domain accuracy. However, obtaining challenging out-of-domain test data is a non-trivial problem. Therefore, we focus on equation recovery for evaluating symbolic regression algorithms for equation discovery. The rationale is that symbolic regression algorithms that perform well in recovering known groundtruth formulas are good candidates to perform well in unknown equation discovery. Existing benchmarks for symbolic regression include equation recovery tasks, however, with only a small number of groundtruth formulas that are publicly known. Moreover, these benchmarks place less emphasis on evaluating the robustness of algorithms in terms of their behavior under changing dimensionality, sampling size, sampling distribution and sampling domain. This, however, is of central importance to practitioners wanting to discover equations for modeling natural phenomena, since data is almost certainly noisy and comes from diverse domains, distributions, and sample sizes. To fill this gap, we introduce the Equation Recovery Benchmark (ERBench), a new evaluation framework designed to rigorously assess algorithms explicitly targeting the task of equation discovery.

Abstract

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