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Towards Diverse Scientific Hypothesis Search with Large Language Models

Haorui Wang, Parshin Shojaee, Kazem Meidani, Kunyang Sun, José Miguel Hernández-Lobato, Teresa Head-Gordon, Jiajun He, Chandan K. Reddy, Chao Zhang, Yuanqi Du

arXiv:2606.10587v1cs.LGcs.AI

TL;DR

Scientific hypothesis search needs diverse high-quality alternatives because validation is noisy, expensive, and underdetermined. EvoDiverse uses parallel-tempered evolutionary search to improve exploration, and across discovery domains it improves quality and diversity under the same validation budget.

  • Problem

    Scientific hypothesis search must produce diverse, high-quality alternatives because validation is noisy, expensive, and underdetermined.

  • Method

    EvoDiverse uses evolutionary searches at multiple temperatures with principled information exchange to balance exploration and refinement.

  • Results

    Across molecular and equation discovery, EvoDiverse consistently improves diversity while maintaining comparable or superior hypothesis quality under the same validation budget.

  • Takeaways & Limitations

    Diversity yields advantages for LLM-based scientific hypothesis search when evaluations are unknown.

  • Takeaways & Limitations

    Multiple temperature pools reduce evaluations available to each pool, leaving pool count, temperature gap, and swap frequency as task-specific hyperparameters.

Abstract

from arXiv · show

Large language models (LLMs) are on the rise for accelerating scientific discovery, most recently in advanced tasks such as generating valid scientific hypotheses. Yet in many discovery settings, the goal is not to identify a single best hypothesis since validation can be noisy and expensive, and scientists benefit from a set of high-quality alternative hypotheses that hedge against downstream uncertainty for the best solutions. Nevertheless, commonly used evolutionary search recipes tend to prioritize optimization over exploration in hypothesis generation, and the resulting selection pressure during the search process leads to diversity collapse. Motivated by these limitations, we formulate hypothesis search as a sampling problem, where the objective is to efficiently produce diverse, high-quality hypotheses under a fixed validation budget. Building on this perspective, we propose \ours, an evolutionary framework inspired by the classical parallel tempering algorithm that searches hypotheses at multiple temperature levels and enables principled information exchange across temperatures to improve exploration without disrupting convergence. Across domains including molecular discovery, equation discovery, and algorithm discovery, our approach consistently improves both hypothesis quality and diversity under the same validation budget, and produces candidates that remain robust under more expensive downstream computational validations.

1. Introduction

Scientific hypothesis search should produce diverse high-quality alternatives rather than optimize for a single best hypothesis. The paper therefore reframes search as approximate sampling and proposes EvoDiverse, a parallel-tempered evolutionary framework inspired by parallel tempering.

  • Motivation: Scientific discovery is stochastic and underdetermined, so multiple competitive hypotheses may remain plausible and diversity is a key search desideratum alongside quality.Approximate simulations, expensive experiments, and hierarchical validation make reliance on one best hypothesis risky.
  • Problem: Existing LLM-based evolutionary search pipelines prioritize optimization over exploration, repeatedly selecting top-ranked hypotheses and thereby underemphasizing diversity.These pipelines use LLMs as evolutionary operators, score proposed hypotheses, and select the highest-ranked candidates until convergence or budget exhaustion.
  • Sampling perspective: The paper formulates hypothesis search as sampling competing hypotheses from a diverse population with probabilities corresponding to their quality.Because exact LLM sampling is challenging and often unsuitable, the paper adopts an approximate sampling perspective rather than enforcing exact sampling.
  • Method: EvoDiverse approximates sampling from a Boltzmann distribution with an evolving power factor through evolutionary search.This formulation retains the sampling perspective without requiring exact sampling from the LLM proposal distribution.
  • Method: EvoDiverse is a parallel-tempered evolutionary framework inspired by the classical parallel tempering sampling algorithm.The framework is introduced to advance diverse scientific hypothesis search while retaining evolutionary search.

2. Background

The background frames LLM-augmented evolutionary algorithms as population-based black-box optimization methods and introduces parallel tempering as a multimodal sampling strategy using multiple temperature levels.

  • Evolutionary Algorithms: Evolutionary algorithms maintain candidate populations that evolve through parent sampling, mating, and mutation or crossover in derivative-free black-box optimization.LLM-augmented variants use LLMs to instantiate the mating operator, proposing offspring whose objective scores guide selection.
  • Boltzmann Sampling: Boltzmann sampling connects sampling and optimization because taking β →∞ recovers optimization behavior when energy is replaced by an arbitrary objective function.Here, β is the inverse temperature parameter and U(·) is the potential energy function.
  • Parallel Tempering: Parallel tempering samples multimodal distributions by mixing a joint distribution over Boltzmann distributions at multiple temperature levels.It combines local exploration within each temperature level with global exploration through communication across levels.
  • Parallel Tempering: Parallel tempering’s joint stationary distribution is the product p(X) = p(x1, x2) := p1(x1)p2(x2), redistributing samples among appropriate intermediate Boltzmann-weighted states.This defines the communication structure across temperature levels.

3. Methodology

The methodology views evolutionary search as approximate sampling and introduces parallel-tempered populations with temperature-dependent selection and Metropolis-Hastings swaps to balance convergence and exploration. This framework preserves temperature-specific distributions while exchanging candidates across levels.

  • Motivation: Evolutionary populations approximately follow a Boltzmann distribution whose changing selection strength controls the trade-off between convergence and exploration.Faster increases in selection strength encourage convergence but reduce exploration, whereas slower increases improve coverage but may fail to converge within a limited budget.
  • Temperature ladder: The method maintains populations across a ladder of temperatures, using tolerant selection at high temperatures and harsher selection at low temperatures.Higher temperatures encourage aggressive exploration, while lower temperatures focus on conservative refinement and convergence.
  • Temperature-dependent selection: A stochastic selection rule tunes exploration through β: higher temperatures use smaller β, while lower temperatures use larger β.As β →∞, selection becomes deterministic; as β →0, selection pressure disappears.
  • Information exchange: Candidate exchange uses Metropolis-Hastings swaps to preserve the product of the temperature-specific target distributions.The operator proposes swapping temperature assignments and accepts the swap with probability A = min{1, a}.
  • Convergence alignment: Because ξ(n) is unknown and implementation-dependent, the method treats it as a dynamic hyperparameter and adjusts it using recent swap rates.The heuristic aims to keep the swap rate within a roughly constant, well-behaved range, such as 30%- 50%.

4. Experiment

Across molecular, equation, and algorithm discovery, EvoDiverse is evaluated under matched validation budgets for hypothesis quality and diversity. Results show that its temperature-based exchange improves exploration and convergence while preserving or increasing solution quality.

  • Experiment: Across molecular, equation, and algorithm discovery, experiments evaluate generated solutions for quality and diversity under the same evaluation budget.The study combines comprehensive quantitative and qualitative evaluations across the three discovery settings.
  • Molecular discovery: In molecular discovery, EvoDiverse reaches high candidate scores quickly while retaining a high volume of diverse samples on JNK3 and GSK3β.It sacrifices only a minimal portion of diversity during early optimization and maintains high QED and SA scores without explicitly optimizing them.
  • Molecular discovery: EvoDiverse exhibits the fastest convergence across molecular targets by Top-10 AUC, whereas Ensemble preserves variety but often remains in low-score regions.The hot pool and swapping algorithm enable faster progress, while the GraphGA-adapted version further improves sample efficiency.
  • Equation discovery: In equation discovery, EvoDiverse consistently achieves substantially higher diversity than baselines while maintaining comparable or often superior best-score and Top-10 performance.Its added diversity is productive, enabling exploration of more informative regions of the symbolic-program hypothesis space.
  • Algorithm discovery: In algorithm discovery, EvoDiverse achieves the best optimization performance across methods while controlled thermodynamic exchange balances exploration and exploitation.Island has the lowest diversity, whereas Ensemble maintains high diversity but weak final performance; EvoDiverse continues improving while baselines plateau earlier.

5. Related Work

Prior work uses LLMs to propose scientific hypotheses that external oracles score, spanning multiple discovery domains. Other approaches have LLMs refine their own hypotheses or interact with human experts.

  • LLMs for Scientific Hypothesis Search: LLM-guided hypothesis search commonly treats language models as heuristics that propose candidates while an external oracle scores them.FunSearch exemplifies this approach for mathematical programs.
  • LLMs for Scientific Hypothesis Search: This evolutionary-style paradigm has been applied to symbolic regression, molecular design, materials discovery, protein optimization, and retrosynthesis planning.The cited works extend the paradigm across mathematical programs and several scientific discovery settings.
  • LLMs for Scientific Hypothesis Search: Complementary approaches have LLMs iteratively analyze and refine their own hypotheses or interact with human experts.These methods broaden hypothesis search beyond external-oracle evaluation.

6. Conclusion, Limitation and Future Works

The paper frames scientific hypothesis search as sampling and proposes a parallel-tempered evolutionary framework to improve candidate diversity. It reports effectiveness across discovery scenarios while identifying budget, calibration, and real-world validation limitations.

  • Conclusion: The framework views LLM-based scientific hypothesis search from a sampling perspective and uses parallel tempering to explore diverse candidates.The paper advocates exploring a wide range of candidates rather than focusing narrowly on a single solution.
  • Conclusion: Quantitative and qualitative validations show effectiveness across multiple scientific discovery scenarios, with diversity helping when evaluations are unknown.These findings support producing diverse alternatives under uncertain downstream evaluation.
  • Limitations: Multiple temperature pools improve exploration and mitigate diversity collapse but reduce evaluations per pool under a fixed oracle budget.The number of pools, temperature gap, and swap frequency therefore require task-specific tuning.
  • Limitations: EvoDiverse is approximate because LLM-induced populations lack known stationary distributions and swap rules can be sensitive to objective scale.Equation discovery requires a log-MSE energy transformation for stability, while automating energy and ladder choices remains future work.
  • Future Works: Deployment requires understanding search spaces and objectives to set the Boltzmann constant, while real-world experiments remain necessary to validate generated hypotheses.Future work also includes improving sample diversity in evolutionary and other commonly used search algorithms.

Impact Statement … B.2.2. IMPLEMENTATION DETAILS

The appendix specifies EvoDiverse’s parallel-tempered evolutionary implementation for molecular discovery, including domain-specific sampling, matched evaluation budgets, task constraints, and baseline configurations. It details how cold and hot pools, survivor selection, swaps, and diversity-aware evaluation are instantiated under a 10,000-call molecular-search budget.

  • Impact Statement: The paper reports no societal impacts requiring specific discussion.
  • A. Pseudocodes: Algorithm 2 uses temperature-specific parent sampling, LLM mating, survivor sampling, and periodic PT swaps to evolve multiple pools.The generic template takes initial pools, temperatures, pool sizes, offspring and parent counts, swap parameters, and iteration controls as inputs.
  • B.1. Domain-Specific Implementation Conventions: Domain implementations differ because task scores, numerical scales, and inherited baseline pipelines differ, while all use two selection stages.The stages construct a mating pool and then produce the next generation; pool i has inverse temperature β_i and temperature T_i = 1/β_i.
  • B.1. Domain-Specific Implementation Conventions: Molecular discovery applies tempering during survivor selection: mating uses raw oracle scores, while survivor sampling becomes sharper in the cold pool and flatter in the hot pool.A small additive constant improves numerical stability and prevents very low-score candidates from receiving zero sampling probability.
  • B.1.1. PER-GENERATION BUDGET DETAILS: Across methods, multi-pool oracle budgets are split so total oracle calls match single-pool baselines, and PT swaps consume no additional oracle evaluations.Per-generation parent and offspring counts are summarized in Appendix Table 4.
  • B.2.1. TASK SPECIFICATION: Molecular search maximizes oracle scores over chemically valid molecules under a fixed budget, rejecting invalid proposals without counting them as evaluations.The JNK3 and GSK3β benchmarks are single-objective black-box searches in a discrete chemical space.
  • B.2.1. TASK SPECIFICATION: Diversity-aware Top-10 selection ranks candidates by oracle score and admits a molecule only when its average Morgan-fingerprint Tanimoto similarity to selected molecules is below 0.4.Diversity-aware Top-10 AUC measures discovery efficiency across iterations, while Diversity-aware Top-10 Avg. averages the final five pre-convergence iterations.
  • B.2.2. IMPLEMENTATION DETAILS: EvoDiverse uses β1 = 0.8 and β2 = 0.2, generates 70 offspring per pool, performs PT swaps every tswap = 5 iterations, and uses B = 10,000 oracle calls.The cold pool emphasizes exploitation, the hot pool encourages exploration, and survivor selection retains the top 3 candidates as elites before tempered sampling.

B.2.3. ADDITIONAL RESULTS AND ANALYSIS

Additional analyses show that EvoDiverse converges faster and generally reaches higher molecular-discovery scores than MOLLEO and other baselines, while also improving classical GraphGA optimization under diversity-aware and score-filtered evaluations.

  • Molecular discovery trajectories: EvoDiverse converges faster to better GSK3β scores than MOLLEO, with only a small loss in diversity.The same analysis reports faster convergence and generally superior scores compared with baselines on the plotted molecular-discovery trajectories.
  • Molecular discovery trajectories: EvoDiverse’s optimization trajectories generally achieve superior average GSK3β and JNK3 scores compared with baselines.Figure 5 averages scores over diversity-aware Top-10 molecules from each iteration.
  • Classical algorithm enhancement: EvoDiverse enhances GraphGA by converging more rapidly to high-scoring molecular samples than baseline methods.The adapted GraphGA version is distinct from the LLM-based EvoDiverse in the main experiments.
  • Classical algorithm enhancement: QED > 0.5 and SA < 5.5 filtering produces the same consistent performance trend as the LLM-based results.The filtered trajectory averages use all molecules from each iteration satisfying both criteria.

B.2.4. ABLATION: METROPOLIS-HASTINGS SWAP … B.3.4. DIVERSITY EVALUATION

The appendices show that EvoDiverse’s Metropolis-Hastings communication improves exploration and convergence across molecular and equation discovery while preserving computational efficiency and candidate quality. Its temperature-specific dynamics produce differentiated, partially overlapping populations and strong diversity under semantic and scaffold-level evaluations.

  • B.2.4. ABLATION: METROPOLIS-HASTINGS SWAP: B.2.4: Both Random Swap and EvoDiverse outperform swap-free Ensemble, while the MH acceptance gate further improves the molecular-search results.Random Swap accepts all proposed swaps unconditionally, whereas EvoDiverse uses the MH gate; results are averaged over 3 seeds.
  • B.2.7. NON–DIVERSITY-AWARE TOP-k SCORES: B.2.7–B.2.8: EvoDiverse balances high oracle score, QED, SA, and scaffold diversity, with at least 10 distinct-scaffold molecules produced by every method at convergence.The chemical-quality analysis reports no sign of reward hacking, and EvoDiverse maintains high average oracle score and high scaffold diversity simultaneously on both targets.
  • B.2.8. SCAFFOLD-LEVEL DIVERSITY AND UNIDOCK DOCKING: B.2.8–B.2.9: EvoDiverse achieves the best JNK3 UniDock score in both evaluated windows and the best GSK3β last-5-iterations score, while its JNK3 Top-10 AUC gains are significant over MOLLEO and Tempering.Its docking variance is also lowest; on JNK3, paired one-sided tests report p = 0.011 versus MOLLEO and p = 0.006 versus Tempering, but p = 0.111 versus Ensemble.
  • B.2.10. DIAGNOSTIC CURVES: B.2.10: The swap acceptance rate stabilizes near 30–40%, while the adaptive powering factor ξ increases monotonically as score distributions sharpen.This qualitative diagnostic behavior matches the corresponding equation- and algorithm-discovery curves.
  • B.3.1. TASK SPECIFICATION: B.3.1–B.3.3: Equation discovery searches discrete symbolic programs for numerically faithful, simple, semantically valid laws, using parallel pools and selective swaps to transfer low-MSE programs while escaping stagnation.The evaluation uses LLM-SRBench with standardized splits and open or closed backbones; EvoDiverse uses β1 = 0.8 and β2 = 0.01, with swaps attempted approximately every 5 iterations over 1000 evaluations.
  • B.3.4. DIVERSITY EVALUATION: B.3.4: Equation-program diversity is measured by mean pairwise CodeBERT embedding distance across pools, capturing structural and functional differences in symbolic programs.Table 15 evaluates whole-buffer and top-100 diversity across datasets and the deepseek-v3.2 and GPT-5 backbones.
  • B.3.4. DIVERSITY EVALUATION: B.3.4: EvoDiverse forms partially overlapping cold- and hot-pool clusters, unlike Ensemble’s disconnected clusters, with overlap representing solutions transferred through thermodynamic swaps.The cold pool concentrates in high-score regions, whereas the hot pool explores a broader portion of the solution space.

B.3.5. ADDITIONAL RESULTS

EvoDiverse consistently improves hypothesis quality and diversity across LLM-SRBench domains, outperforming prior LLM-based approaches across backbone models. Extended comparisons show gains against both LLM-based baselines and the non-LLM symbolic regression method PySR.

  • Extended comparisons: Appendix Figure 9 extends the comparison to Tempering, Ensemble/LLM-SR, and PySR, with LLM-based methods averaged over DeepSeek-V3.2 and GPT-5.The comparison covers the four LLM-SRBench domains.
  • Extended comparisons: EvoDiverse shows substantial gains across all four LLM-SRBench domains despite PySR’s strong performance as a non-LLM symbolic regressor.LLM-based results are averaged over DeepSeek-V3.2 and GPT-5 backbones to highlight method-level trends independent of backbone choice.
  • Quality comparison: Appendix Table 16 compares EvoDiverse with Tempering and Ensemble/LLM-SR using Best NMSE, Top-10 NMSE, Best Acc0.1, and Top-10 Acc0.1 across all LLM-SRBench domains.Evaluations are separated by DeepSeek-V3.2 and GPT-5; NMSE metrics are lower-is-better, while accuracy metrics use threshold 0.1 and are higher-is-better.
  • Quality and diversity: EvoDiverse improves both best-case solution quality and the quality and diversity of the elite selected hypothesis set across domains and backbone models.These results are reported in Table 2 and Appendix Table 23, emphasizing exploration of better and higher-quality regions of the hypothesis space.

B.3.6. ABLATION: ENERGY FUNCTION CHOICE FOR PT SWAP

The ablation isolates EvoDiverse’s PT swap energy function, comparing log-MSE and raw-MSE acceptance while keeping selection and other settings fixed. Log-MSE is motivated by MSE’s wide dynamic range and supports useful cross-temperature coupling where raw-score acceptance can become exponentially unlikely.

  • Energy-function design: EvoDiverse uses h(x) = log MSE(x) for PT swaps but raw MSE for parent-selection Boltzmann weights.MSE values can span 10^-8 to 10^1 within one symbolic-regression run, making raw differences unstable across error regimes.
  • Ablation setup: The ablation replaces the swap criterion with raw-score differences while keeping selection, adaptive ξ, budget, and hyperparameters identical.Only EvoDiverse is affected because Ensemble and Tempering have no swap step.
  • Ablation findings: Raw-score swaps accept hot-to-cold exchanges only when the hot candidate is essentially as good as the current cold candidate.When |s_b − s_a| is not small, exponentially small acceptance probabilities prevent adaptive ξ from recovering useful coupling across temperatures.
  • Ablation findings: The log-MSE swap instead compares error on an order-of-magnitude scale while retaining the Metropolis acceptance step with adaptive scaling.The swap continues to preferentially move low-energy candidates from hot pools toward cold pools.
  • Experimental control: Both variants use p(x) ∝ exp(−β MSE(x)), so the comparison isolates the effect of the swap energy function.The ablation evaluates EvoDiverse on BIOPOPGROWTH using 24 problems, a DeepSeek-V3.2 backbone, and B=1000 evaluations.

B.3.7. ABLATION: METROPOLIS-HASTINGS SWAP … B.4. Algorithm Discovery

The ablations show that EvoDiverse’s Metropolis-Hastings swap gate, temperature settings, and multi-pool design support equation-discovery performance and diversity. Improvements are highly significant, while diagnostic curves show acceptance stabilizing near the target across domains and backbones.

  • B.3.7. ABLATION: METROPOLIS-HASTINGS SWAP: Both swap-enabled methods improve over Ensemble, while the Metropolis-Hastings gate delivers the strongest improvement.The comparison uses an unconditional Random Swap variant and an Ensemble baseline without swaps.
  • B.3.8. HYPERPARAMETER SENSITIVITY: Performance is stable for β1 ∈{0.8, 1.0} paired with β2 ≤0.1, degrading only at extreme temperature settings.β1 = 1.5 collapses pool diversity, while β2 = 0.5 removes the temperature gap between pools.
  • B.3.8. HYPERPARAMETER SENSITIVITY: A swap period of 2–5 works well, whereas longer periods reduce diversity.The sensitivity analysis also varies the per-swap batch size k.
  • B.3.9. GENERALIZATION TO MORE TEMPERATURE LEVELS: The three-temperature variant marginally improves both diversity and accuracy over the two-temperature default.It uses β ∈{0.8, 0.2, 0.01} on the Physics subset, confirming generalization to more pools.
  • B.3.10. STATISTICAL SIGNIFICANCE: Both EvoDiverse improvements over the Tempering and Ensemble baselines are highly significant (p < 0.001).The result comes from paired one-sided t-tests on equation discovery.
  • B.3.11. DIAGNOSTIC CURVES: The acceptance rate stabilises rapidly near the target (∼30%), while ξ increases monotonically as score distributions sharpen during search.This behavior appears across four LLM-SRBENCH domains and both LLM backbones.
  • B.4. Algorithm Discovery: The qualitative diagnostic behavior matches that observed for molecular and algorithm discovery.The comparison is reported through Appendix Figures 8 and 14.

B.4.1. TASK SPECIFICATION AND IMPLEMENTATION DETAILS · B.4.2. DETAILED DIVERSITY METRICS · B.4.3. DIVERSITY LANDSCAPE VISUALIZATION

The circle-packing experiments define a constrained program-search task and evaluate methods under matched LLM prompting and a 1,000-program budget. Diversity analyses show that EvoDiverse differentiates pools while preserving exchange, unlike baseline homogenization or asymmetric contributions.

  • B.4.1. TASK SPECIFICATION AND IMPLEMENTATION DETAILS: The task places n = 26 non-overlapping circles inside a unit square, maximizing the sum of radii through Python programs.Programs are validated against geometric constraints and follow the standard formulation from prior work.
  • B.4.1. TASK SPECIFICATION AND IMPLEMENTATION DETAILS: All methods use DeepSeek-V3.2, identical prompting, fitness-proportional solution sampling, and 1,000 total evaluations.The matched budget is also used for baseline configurations.
  • B.4.1. TASK SPECIFICATION AND IMPLEMENTATION DETAILS: EvoDiverse uses cold and hot pools with β = 1.0 and β = 0.25, swaps every 5 iterations, and adaptive energy scaling targeting a 0.3 ± 0.1 swap rate.Mating uses pool-specific Boltzmann sampling, while updates append the child and remove the worst-scoring candidate at capacity 500.
  • B.4.2. DETAILED DIVERSITY METRICS: Table 23 combines TF-IDF and CodeBERT diversity measures with per-pool analysis and elite-set contributions.EvoDiverse’s Cold pool contributes 68% of elites, whereas Island EA contributes 50-50 and Ensemble EA shows catastrophic asymmetry.
  • B.4.3. DIVERSITY LANDSCAPE VISUALIZATION: PCA visualizations use TF-IDF embeddings and separately encode pool identity and program scores to relate diversity to optimization quality.The panels compare EvoDiverse’s Cold/Hot pools with baseline Pool 0/Pool 1 structures.
  • B.4.3. DIVERSITY LANDSCAPE VISUALIZATION: EvoDiverse forms partially overlapping clusters: the Cold pool concentrates in high-score regions, while the Hot pool explores a broader solution-space area.The overlap represents solutions transferred between pools through thermodynamic swaps.
  • B.4.3. DIVERSITY LANDSCAPE VISUALIZATION: Island EA forms a merged, color-mixed cluster because migration synchronizes both pools into an identical population.Both pools occupy the same region with similar score distributions, indicating homogenization.

B.4.4. ABLATION: METROPOLIS-HASTINGS SWAP … C. Prompts

Across ablations and diagnostics, EvoDiverse preserves quality and diversity advantages under matched evaluation budgets, while its prompts operationalize exploration and improvement across molecular, equation, and algorithm discovery. The circle-packing results further show differentiated populations, adaptive swap behavior, and a structured high-performing solution.

  • B.4.4. ABLATION: METROPOLIS-HASTINGS SWAP: Both Random Swap and EvoDiverse improve over the no-swap Ensemble baseline, while diversity values are recomputed over unique evolution-trace programs and preserve the methods’ relative ranking.Random Swap disables the MH gate and accepts all proposed swaps unconditionally, whereas EvoDiverse retains the MH acceptance mechanism.
  • B.4.5. COMPUTATIONAL COST: Under the same oracle budget of 1,000 program evaluations, methods use comparable total tokens and have only modestly varying median iteration times.Reported total token usage is 11–14 M and median iteration time is 20–39 s; variation reflects program length and server load.
  • B.4.6. DIAGNOSTIC CURVES: EvoDiverse’s powering factor ξ adapts monotonically to keep empirical swap acceptance near the 30–40% target.The diagnostic is reported for circle packing and is consistent with analogous molecular- and equation-discovery diagnostics.
  • B.4.7. T-SNE VISUALIZATION: The t-SNE view reproduces the PCA patterns: EvoDiverse has differentiated but overlapping cold and hot clusters, unlike Island’s homogenized and Ensemble’s disconnected populations.The disconnected Ensemble clusters fail to share information.
  • B.4.8. DISCOVERED SOLUTION: EvoDiverse discovers a circle packing with n = 26 and sum of radii = 2.5461, using corner, edge, and interior placements for structured space utilization.The configuration contains 4 corner circles, 12 edge circles, and 10 interior circles, with varying sizes and boundary-aware placement.
  • C. Prompts: In molecular discovery, hot-pool prompts ask for novelty while maintaining competitive JNK3 scores, whereas cold-pool prompts prioritize higher target scores through crossover, mutation, or new proposals.Both prompts require a rationale and a final molecule in a specified SMILES format.
  • C. Prompts: Equation-discovery prompts provide function skeletons, scored program examples, and successive improved versions for force and energy tasks.The examples specify inputs, parameters, outputs, and candidate equations for multi-variable functions.
  • C. Prompts: Algorithm-discovery prompts support full rewrites and targeted diff-based improvements while explicitly valuing both fitness and diversity across feature dimensions.Full rewrites must preserve inputs and outputs, while diff-based prompts require exact SEARCH/REPLACE formatting and prohibit rewriting the entire program.
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