Helix: Evolutionary Reinforcement Learning for Open-Ended Scientific Problem Solving
Authors: Chang Su
Zhongkai Hao
Zhizhou Zhang
Zeyu Xia
Youjia Wu
Hang Su
Jun Zhu
Venue: ICLR 2026 Poster (2026)
Keywords: Large Language Models
Reinforcement Learning
Evolution Strategies
Scientific Discovery
Primary area: foundation or frontier models, including LLMs
TL;DR: We introduce HELIX, a Hierarchical Evolutionary Reinforcement Learning framework with In-context eXperiences, achieving superior performance over GPT-4o pipelines on open-ended scientific tasks

Abstract: Large language models (LLMs) with reasoning abilities have demonstrated 
growing promise for tackling complex scientific problems. Yet such tasks are inherently domain-specific, unbounded and open-ended, demanding exploration across vast and flexible solution spaces. Existing approaches, whether purely learning-based or reliant on carefully designed workflows, often suffer from limited exploration efficiency and poor generalization. 
To overcome these challenges, we present **HELIX**---a 
**H**ierarchical **E**volutionary reinforcement **L**earning framework with **I**n-context e**X**periences. HELIX introduces two key novelties: (i) a diverse yet high-quality pool of candidate solutions that broadens exploration through in-context learning, and (ii) reinforcement learning for iterative policy refinement that progressively elevates solution quality. This synergy enables the discovery of more advanced solutions. On the circle packing task, HELIX achieves state-of-the-art result with a sum of radii of 2.63598308 using only a 14B model. Across standard machine learning benchmarks, HELIX further surpasses GPT-4o with a carefully engineered pipeline, delivering an average F1 improvement of 5.95 points on the Adult and Bank Marketing datasets.

Published as a conference paper at ICLR 2026
HELIX: EVOLUTIONARY REINFORCEMENT LEARNING FOR OPEN-ENDED SCIENTIFIC PROBLEM SOLVING
Chang Su1.2.1, Zhongkai Hao Zhizhou Zhang', Zeyu Xia?.1 Youjia Wu', Hang Su²,
Jun 2hu
Bosch (China) Investment Co., Ltd., "Tsinghua-Bosch Joint Center for ML, Tsinghua University, & Dept. of EE, Tsinghua University,
Dept. of Comp. Sci. & Techn., Tsinghua University
(su-c24, hzj21, xia-zy25 emma
em t...siinght edu. cn,
[dcsz), suhangss sttsinghua edu.. en, en. (Zhizhou. zhang, coujia.wufeace www.bbscch...oo.. com
ABSTRACT
Large language models (LLMs) with reasoning abilities have demonstrated grow- ing promise for tackling complex scientific problems. Yet tasks are inherently domain-specific, unbounded and open-ended, demanding exploration across vast and flexible solution spaces. Existing approaches, whether purely learning-based or reliant on carefully designed workflows, often suffer from limited exploration efficiency and poor generalization. To overcome these challenges, we present HE- LIX - Hierarchical Evolutionary reinforcement Learning framework with In- context eXperiences. HELIX introduces two key novelties: (i) a diverse yet high- quality pool of candidate solutions that broadens exploration through in-context learning, and (ii) reinforcement learning for iterative policy refinement that pro- gressively elevates solution quality. This synergy enables the discovery of more advanced solutions. On the circle packing task, HELIX achieves state-of-the-art result with sum of radii of 2.6 2.63598308 using only 14B model. Across standard machine learning benchmarks, HELIX further surpasses GPT-4o with a carefully engineered pipeline, delivering average Fl improvement of 5.95 points the Adult and Bank Marketing datasets.
INTRODUCTION
Solving complex scientific problems with large language models (LLMs) is an important and creasingly active research direction (Forootami (205). By leveraging and enhancing their reasoning capabilities, LLMs have demonstrated promising results in tackling challenging scientific tasks, such as symbolic regression (Shojaee et al., 2022),, molecular generation (Liu et al. (2024), and diffi- cult mathematical optimization problems (Ahmed & Choudhury 2024). Addressing such problems holds the potential to advance the boundaries of human knowledge and reshape scientific discovery.
While LLMs have shown promising applications, complex scientific problems remain particularly challenging due to three intrinsic characteristics. First, they are domain-specifie, with unique envi- ronments and
constraints that differ across various tasks. Second, they are open- ended, requiring exploration of vast and flexible solution spaces. Third, they unbounded, often with no known or guaranteed global optimum.
To address these challenges, argue that powerful LLM for solving complex scientific prob- lems must possess three corresponding key abilities: (1) learning from experience, i.e., it should enable task-specific policy adaptation by incorporating feedback from previous trials, addressing the domain-specific nature of each problem. (2) Balancing quality and diversity, i.e., it should maintain diverse population thoroughly explore the vast and flexible solution spaces inherent in
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coils and core
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(a) Housing Dataset (Finding ML algorithms)
(b) Beam and Inductor Design (Finding better geometric shape)
Figure 1: The figure demonstrates how framework progressively discovers new insights and refines solutions over iterations. (a): Reward curve for the housing dataset optimization, where improvements are achieved through iterative adoption of better models, parameter tuning, and fea- ture engineering, with the final reward of 1,758 corresponding to an RMSE of 1.747. (b): Reward curves for the beam and inductor design tasks, where the algorithm explores novel geometries and combines favorable structural features to enhance performance.
open-ended tasks. (3) Exploration based on the shoulder of giants, i.e., it should iteratively build upon existing high-quality solutions to extend the known limits of unbounded problems.
However, recent works largely lack the capabilities outlined above, limiting their effectiveness on complex scientific problems. Existing approaches fall into two categories. Post-training methods SFT, RLVR) fine-tune LLMs on domain-specific datasets, as in AlphaCode (Li et al. 2022) and Deepseek-RI (Ren et 2025), achieving strong results in code generation and mathematical rea- soning. Yet such methods often suffer from entropy collapse (Cui et al.) 2025) and. as shown in Yue al. (2025), rarely move beyond the base model's capabilities. This makes it difficult discover fundamentally new solutions, especially when sparse rewards further limit exploration. Workflow- driven approaches embed I.LMs in predefined pipelines to improve task-specific performance. Ex- amples include integrating genetic algorithms with L.LMs for enzyme discovery (Nana Teukam et al.) 2025), establishing LLM-driven evolutionary loops such as LLaMEA (yan Stein & Bäck 2024), or applying evolutionary strategies prompts (Agrawal et al. 2025). While effective narrow tasks, these systems are highly sensitive to workflow design and rely on static pretrained knowledge, mak- ing hard to reuse past discoveries to guide iterative search. Both categories thus struggle gen- eralize in open-ended scientific domains where efficient exploration and continual refinement are essential.
To this end, we propose HELIX- Hierarchical Evolutionary reinforcement Learning framework with In-context eXperiences. First, learn from experience, HELIX updates the LLM policy us- ing reward signals by reinforcement learning to progressively improve solution quality. Meanwhile candidate solutions explored by the model forms a population for evolving algorithms. Secondly, to balance the quality and diversity, we propose to rank and select samples using both diversity and reward, inspired by a classic multi-objective evolutionary algorithm named ASGA-III(ed et al.) 2002). Specifically, to better measure the novelty of solution, we compute diversity using pre- trained language embedding model and estimate the diversity by K-Nearest Neighbors. Finally, we enable the model to stand the shoulder of giants by adding prompt constructed by the best solu- tions in the population to guide the model to generate new solutions. By using the in-context learning paradigm, seamlessly unify and integrate evolutionary learning with reinforcement learning to explore the vast solution space in complex scientific problems.
In experiments, evaluated HELIX 20 tasks across five diverse categories. Compared with strong task-specific baselines and advanced proprietary models such as GPT-4o, HELIX achieves superior performance 17 tasks, demonstrating its ability to iteratively refine solutions and update
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its policy towards better results. Further analysis via ablation studies confirms that each component of HELIX contributes critically to performance. Notably, success on these unbounded and open- ended tasks suggests that iterative, diversity-aware exploration can provide useful insights for other scientific and engineering problems.
RELATED WORK
Reinforcement learning of LLMs. Training LLMs or LLM-based agents with reinforcement learning (RL) has recently attracted significant attention. This includes reinforcement learning from human feedback (RLHF) to align models with human preferences, as well as RI. with veri- fiable rewards (RLVR) enhance reasoning. mathematical problem-solving, and coding capabil- ities. Beyond improving reasoning, RLVR-style training can also elicit new capabilities such as tool use (Feng et al. 2025) and information retrieval (Jin et al., 2025). A representative method is GRPO (Shao et al. 2024h, which normalizes rewards within groups of samples. Variants such as DAPO (Yu et al. (025) and Dr.GRPO (Liu et al.) 2025) further improve GRPO through refined data sampling strategies and advantage estimation techniques. While RL can improve generalization in specific domains, the training process often suffers from decreasing entropy and diversity over time, hindering effective exploration. Some approaches, such KL-Cov (Cui al., 2025), ) attempt address this limitation by applying KL penalty solely to tokens with high covariance to preserve entropy. However, for complex scientific problems, these memory-less RL methods- the sampling context for the same problem remains fixed wred-struugle to leverage solutions that have al- ready been discovered, making it difficult to build upon prior explorations.
Evolutionary algorithms. Evolutionary algorithms are a classic approach for tackling complex optimization problems. They use "gene" represent solution for the problem and use random mutation to explore the whole solution space. Recently, AlphaEvolve (Novikov et al.) (025)) treats code as the "gene" and applies I.LM-driven mutations, successfully integrating LLM agents with evolutionary algorihhms-openii many works have adopted similar agent-based workflows to address scientific tasks such as CUDA code optimization (Lange al. 2025), drug discovery (Gao et al.) 22025). and complex scientific software usage (Fan et al. [025 Pham et al. 2025). However, such methods typically require highly problem-specific workflow logic and prompt design, which greatly limit their effectiveness in solving more general and complex problems.
the door to solving complex scientific problems. Since then,
PROPOSED METHOD
3.1 OVERVIEW
To tackle the challenges of applying large language models (LLMs) to complex scientific discovery tasks, we propose HELIX, a hybrid framework that integrates reinforcement learning with evolu- tionary search. The goal is to enable LLMs to efficiently explore large and flexible solution spaces while maintaining diversity and exploiting previously discovered high-quality solutions. The frame- work is composed of three complementary modules: (1) A reinforcement learning framework that updates the policy parameters based on verifiable reward, allowing the model to learn from experience and progressively improve its reasoning capability. (2) A multi-objective evolution- ary mechanism that balancing solution quality and diversity, ensuring that the population retains both high-performing and diverse candidates for further expansion. (3) An in-context learning mechanism that incorporates multiple past trials into the prompt, enabling the model to build upon previously discovered solutions and expand its exploration
the shoulder of giants.
We consider the task as an optimization problem that has a solution space of code. Let E S denote a candidate solution, represented code written in domain-speciffe language Python, YAML,, or other DSLs). We define an objective reward function R(-) which only depends on the current solution (state). The optimization objective is to find valid E S to maximize the reward:
max R(s). eES
(I)
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Tasks & Solution Pool
Language Model
Lineage Tree
Prompl Construction
Packing
Initial Solution
Modified Solution
S
Samples
Task Description
Py
.....*
23
Foes = T0
S
Current Solution
Physics Simulttion
52
Provide improvements to this solution.
DSL
BEEE
5:
Update Policy
Symbolcc Regrosioon
em B Pyfooo
Evaluation Pipeline
Previous Trials Trials
Equ
NSGA-11 Select
RL Algorithm
Trials
Function Minimization
Update
Samples
Scored
Dataset
Reward
Selestal
Solution
Pando Front
Evaluation Pipeline
Machin: Leaming Tacke
Py
Diversity
Bathday
Funaly
Target
Prezees
Solation
Eubel
Divenity
Caleatir
Exalary
RAs unl
Figure 2: Illustration of HELIX framework. The workflow begins with a dataset containing task descriptions and a pool of initial solutions, which are taken by LLM inputs. The LLM will modify and update the original solution and generate a new one, represented as descendants lineage tree. labeldd solutions will be used update policy
After the evaluation pipeline, the resulting parameters via reinforcement learning. Those samples will also be selected by NSGA-II algorithm to construct promising yet diverse candidate solutions for population evolution.
To explore and search for new solutions, we use an LLM policy aa that iteratively (itaoess) current solutions. Given timestep f, we sample solution s, from Pt, the of candidate solutions at 1-th step. The LLM will output an action ag E AA, which is an edit or modification applied to St, obtain new solution St-1 T(s;. where T is the transition function. Our goal to improve the policy's 's ability to find better solutions. The objective is defined as follows,
max 0
EsSoPee am- g.s.) [R(s az)]
where q is the prompt constructed in equation and R(s az) R(8 ) is the reward of the new solution with a slight abuse of notations. We leverage GRPO (Shao al. 2024). reinforcement learning algorithm, to update LLM policy πι. By maximize the reward in equation 2 the LLM will learn to enhance current solution 8, towards higher reward, which will finally leads to improvement in equation [
To address the exploration-exxloitatton trade-off and prevent entropy collapse in RL, we incorporate evolutionary algorithm in selection of candidate solutions. Suppose D. {s:} } is the set of all solutions generated in the t-th iteration and Do = {5o} is the set of initial solution, the candidate
solution for t-th step can be constructed as
P. Selecr D.).
s=0
where NSGA-II (Deb et al. 2002) is a sample selection strategy widely adopt in evolutionary al- gorithms, which ensures retention of high-reward and diverse candidates. This formulation allows the model to iteratively improve its policy while exploiting previously found high-quality solutions as starting points for further exploration. Figure 2 provides brief summary of our method and the formalized algorithm can be found in Appendix
3.2 POLICY OPTIMIZATION ALIGNED WITH EVOLUTIONARY SEARCH
As the evolutionary process unfolds, updating the model parameters becomes crucial: it enables the policy to learn from both successful and failed trials, generate higher-quality solutions, and dynamically adapt to the shifting input distribution induced by the evolutionary search. Reinforce- ment learning is particularly suitable in this scientific setting, since open-ended scientific tasks lack standard answers and typically provide only sparse reward feedback. Motivated by the design of GRPO (Shao et al. 2024), develop reinforcement learning-based policy update mechanism
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tailored our framework. GRPO has proven effective in enhancing LLM reasoning mathemat- ical and programming tasks (Guo et al. 2023). D and its multi-sample generation naturally provides
diverse reasoning-drinn driven outputs that enrich the evolutionary dataset, making it natural inspiration for our method.
Formally, given prompt q, the model will generate G rollout sequences {ayhessa with policy Taa The GRPO objective is then defined as:
(0) = -Ee Beec { (05]5-1~=an (-14,81)
G
layl
min (r5, clip(rs. +,1+,)AAa - 3DKL(Tel|TTt)
where (0) = malagalgaa,.o
Te (= is the token-level policy ratio, Aj,; R[s memammmaa.... [[(s,.)))
std, (R(
the token-level advantage, 6 the clipping parameter, and / controls the KI. divergence penalty against a reference policy Tzl.
In order to fully leverage the in-context learning ability of LLMs, enabling the model to learn from feedback of previous trials and propose advanced solutions, we construct the prompt in the fol- lowing manner:
q Construc(Propp((p)P[5 R(s;), F(s:))UE)()) R(F(() F(f(*) (s:f(((ss))))))))))))))))))))))) (sk)))) (5)
where p is the problem description, (s)) is the k-th ancestor of 8t in lineage (a historical trace of the solution s, 's iterative refinement), R(-)) represents the reward function and F(.) denotes the auxiliary feedback (e.g., textual structured evaluations) provided by the evaluator to guide future refinements. By constructing prompts using memory of previous feedback and rewards along lineage tree, it ensures the model effectively explores across challenging solution spaces.
3.3 EVOLUTIONARY MECHANISM FOR BALANCING QUALITY AND DIVERSITY
In unbounded scientific research tasks, it is crucial to explore multiple promising ideas or directions. Thus, the optimization process must balance quality, i.e., high-reward solutions that serve as strong starting points for refinement, with diversity, which sustains broad exploration across the solution space. We design the evolutionary search algorithm to a multi-objectiive optimization that natu- rally achieves a trade-off by maintaining a population that simultaneously improves in reward and preserves diverse candidates. Specifically, we innovatively adopt NSGA-I et al, (2002), which a powerfull multi-objective optimization algorithm, to filter high quality and diverse samples on the Pareto front of reward and diversity for subsequent expansion. To further encourage more diverse exploration and enable more accurate diversity computation, we propose to computate the diversity score based on its semantic embedding similarity using a pretrained language embedding model.
Diversity measurement. To quantify the diversity of candidate solutions, first normalize each solution into a canonical code format and encode it into an embedding vector using pretrained embedding model. Let D Ub D. represents the union of all solutions, E(s) RRd denote the embedding of solution s E D. For any solution s;, its diversity score is computed by measuring the average similarity to its k nearest neighbors in the embedding space:
E(s.) - E(s,)
Div(s;) =
Σ jEN..()
(6)
k
[(s
where Ni) denotes the indices of the & nearest neighbors of s; in D. measured by cosine similarity. A higher Div(s:) indicates that is more distinct from other solutions, thereby contributing to population diversity.
NSGA-II based selection. Given both reward score R(s) and diversity score Div(s). each didate solution can be mapped to a wo--dimnsioona objective space. We then adopt the NSGA- II (Deb et al. 2002) algorithm select high-quality and diverse samples. NSGA-II first applies
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nondominated sorting procedure to partition solutions into multiple fronts based on Pareto domi- nance, where solution Sa dominates &, if R(sa) 2 R(s)) and Div(sa) 2 Div(s;) with at least one strict inequality. To further ensure diversity preservation within each front, NSGA-II computes a crowding- crowding-distance measure and selects representative samples that are well spread in the objective space.
By combining nondominated sorting with diversity preservation, the resulting population P retains candidates that both high-reward and diverse. This mechanism allows the model to continuously exploit promising solutions while sustaining exploration across multiple distinct solution trajecto- ries.
4 EXPERIMENT
In this section, we first introduce the experimental setup, including the tasks we selected for bench- marking the model's ability to solve open- scientific problems. Then, we present extensive experiments demonstrating that HELIX effectively enhances model capability, integrates historical experience, and balances reward with diversity, leading to significant improvements over existing baselines in solving unbounded and open-ended scientific challenges. Finally, the ablation studies reveal how different components of the framework work together a complementary manner.
4.1 EXPERIMENT SETTING
Tasks. To comprehensively evaluate the model's capacity for complex scientific reasoning, we design experiments on five representative categories of tasks. These tasks are particularly suited for our study because they are unbounded, lacking guaranteed global optimum, open-ended, requiring exploration over vast and flexible solution spaces and domain- -epeaiieeeecffce, containing unique constraints and complex background Success in these tasks not only demonstrates the model's ability to search beyond local optima, but also provides insights that can inspire solutions in broader scientific and engineering domains.
1. Machine Learning Tasks. We selected three representative datasets: Adult in- come (Becker & Kohavi 1996). Bank marketing (Moro et al. 2014) and Boston hous- ing (Harrison Jr & Rubinfeld, 1978) dataset evaluate the model's ability to solve machine learning tasks. These tasks reflects the open-ended challenge of combining ML algorithms for novel applications, with potential implications for autonomous scientific workflows.
2. Physics Simulation Tasks. These tasks combine geometric structures design and optimiza- tion in multi-physics environments in distinet fields. The design space of these problems has a very high degree of freedom with few global optimal solution.
3. Circle Packing Problems. The objective of these tasks is to maximize the sum of radii of circles packed within given shapes. It allows multiple feasible arrangements and there is proved global optimum solution currently.
4. Function Minimization. requires LLM to write code find the global minimum point given functions. Agents can search freely for new mathematical optimization methods in code space.
5. Symbolic Regression. A benchmark (Shojaee al. 2025 evaluates the ability of LLMs to hypothesize underlying expressions for noisy data. The model needs to search among vast possible expression set and utilize domain specific knowledge to find solution.
Models. We selected the DeepSeek-RR -iitssi-----itil--------- model family for our experiment due to its strong reasoning capabilities and manageable size, which is critical for performing complex sci- entific tasks under computational constraints. Among the model family, the 14B version offers an optimal balance between efficiency and performance, and was selected the model in the main results. For physics simulation tasks that require strong geometric reasoning ability and physical prior knowledge, we utilize the 32B version of the model.
Baselines.
We compare approach against three key baselines:

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Table Results of main experiments. All values correspond to the best outcome obtained across all attempis. We use t to indicate that larger values correspond to better performance, and + represents the opposite. We highlighted the best results each task in bold. "NA" denotes non-convergence or unsuitability for given case.
Task Specilic Methods
Direct Prompt
Open Evolve
Ours
Tasks
LightGBM
RRL
Qwen
GPT-4o
Qwen
GPT-4o
Adult 1
Machine Learning
Bank Marketing
75.28
76.32
0.00
76.91
75.66
78.54
80.65
Boston Housing 1.
3,258
3.966
3.149
3.031
2,937
2,937
1.747
Transpareut Conductors
0.060
NA
0.060
0.059
0.059
0.056
0.049
Tasks
Parameter Scan
Topology Opt
Qwen
GPT-4o
Qwen
GPT-4a
Inductor
6.111
6.248
2,584
0.001
1.637
1.652
9.609
Beam Bending
4,771
NA
5.407
4,005
10.793
6.352
17.298
Physics Simulation
Magnetic Torquet
10.273
NA
0.323
1.201
3.488
1.607
11.045
Periodic Heat f
1,206
NA
1,258
1.255
1.233
1,266
1.278
Demultiplxer
18.322
23,555
3,364
4.532
12.341
8.645
14.260
S1.SQP
Algo
Qwen
GPT.40
Qwen
Circle Packing
Packing in Unit Square 1
2,519
2.345
1.673
1,900
1.586
2,611
2.636
Packing in Unit Disk
4,522
3.896
4.608
3,290
4.604
3,984
4.664
SLSQOP
Trust-constr
Qwen
GPT.40
Qwen
Eggholder
0.705
0.688
1.000
0.959
1.000
1,000
1.000
Mishras Bird
0.814
0.764
1.000
0.996
1.000
1,000
1.000
Function Minimization
Keanes Bump f
0.692
0.886
1,000
1,000
Keanes Bump 20d I
0.603
NA
0.794
0.637
0.596
0.983
1.000
Keanes Bump 30d f
0.594
NA
0.923
0.625
0.677
0.668
0.994
Tasks
M.
LaSR
Qwen
GPT-4o
Qwen
GPT-4o
Cbemistry
4.12e-6
9.11e-5
2.66e-5
2.44e-6
1.59e-5
9.52c-6
7.32c-6
Symbolic Regression
Biology
3.06e-6
1.53e4
1.26c-4
7.52e-5
1.64c4
5.31e-5
2.98e-8
Physics
7.62e-5
9.94c-4
2.71e-4
1.13e4
2.76e-5
1.22e44
2.76e-5
Material Science
9.23e-6
7.146-6
1.85e-6
6.99e-7
1.94e-6
4.46e-6
1. Direct Prompt -Test-Time Scaling): Queries the model directly and selects the best out- come from multiple samples to establish performance upper bound of base model.
2. Open Evolve (Sharma 2025): An open-source implementation of the AlphaB- volve (Novikov et al. 2025) framework, which uses an evolutionary algorithm with multi- ple LLLM roles (e.g., proposing code mutations, evaluating fitness) to iteratively generate, test, and evolve code or solutions across generations.
3. Task-Specific Methods: Represents results from established algorithms designed for each specific problem. Details of these methods can be found in Appendix C
4.2 MAIN RESULTS
Table presents the results of our methods compared various baselines. The best results in each task are highlighted in bold. Since we selected multiple heterogeneous tasks, their evaluation metrics are not the same. The detailed definitions and specific evaluation criteria are deferred to Appendix B
Across the 20 benchmark tasks, our method achieves the best performance on 17 tasks, surpassing all competing baselines. Compared under the same model settings, our framework consistently out- performs Direct Prompting across all benchmarks. Against OpenEvolve- the open-source version of AlphaEvolve-it achieves superior results on 19 tasks. These results clearly highlight the strength of our framework in solving open-ended scientific problems among various domains compared to other approaches.
Notably, we observe that the base Qwen models perform relatively poorly on certain tasks such as Bank Marketing and Magnetic Torque, exhibiting low rewards even in the best of 64 direct trials. However, our framework significantly improves performance these cases by leveraging parameter updates and in-context learning to effectively incorporate feedback from the exploration process.
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1.00
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Epoch
(a) Inductor design
(b) Adult income prediction
Figure 3: Convergence analysis on the Inductor and Adult tasks. The curves show the progressive improvement of average reward and validity during training, demonstrating that our framework ef- fectively leverages reinforcement learning feedback and evolutionary dynamics to produce increas- ingly valid and high-quality solutions.
This demonstrates that our approach can partially overcome the limitations of weaker base models by iteratively evolving toward superior solutions.
To further assess the competitiveness of our approach against state-of-the-art scientific discovery systems, we compared it with GPT-4o, of the most advanced closed-source models. Remarkably, our method outperforms GPT-4o on 18 tasks, regardless of whether GPT-4o is equipped with multi- role collaborative reasoning frameworks. These results highlight that our framework can fully ex- ploit the prior knowledge of smaller models through reinforcement learning, enabling cost-efficient and effective solutions complex scientific problems.
In comparison with task-specific methods, which are typically crafted by human experts for par- ticular domains, our framework still achieves superior performance on 17 tasks. Specifically, in the circle packing task, we establish a new world record 2.63598308 using only 14B model. For the Transparent Conductors dataset, derived from a human- cumaan-prticcpaaioon competition (Ziletti et al.) 2017). our framework attains the second aecond-highest lights its ability to iteratively evolve within open-ended solution spaces and autonomously uncover novel solutions that go beyond manually designed approaches.
the participants' leaderboard. This high-
To provide further evidence that our framework effectively integrates reinforcement learning and evolutionary algorithms, analyze its convergence behavior two representative cases: inductor design and adult income prediction. Figure3]plots the average reward and validity of model outputs during training. Both metrics exhibit clear upward trend: the validity rate rises steadily, showing that the model increasingly generates outputs that satisfy task constraints, while the average reward improves, reflecting higher-quality solutions. This dual improvement demonstrates that reinforce- ment learning progressively strengthens the model's 's intrinsic reasoning ability. It also indicates that the quality of the evolving population keeps improving, enabling the model leverage in-context feedback as well as intuitions from high-reward solutions to generate better outputs.
4.3
ABLATION STUDY
4.3.1
EFFECTIVENESS OF FRAMEWORK COMPONENTS
To better understand the contribution of each component in our framework, we conduct ablation studies on the Boston Housing and Circle Packing tasks. We design several controlled variants by selectively disabling or simplifying parts of the algorithm: TopScore, where only the highest-reward candidate in the dataset is selected for further evolution; TopDiv, where selection relies solely on diversity without considering reward; Random, where candidates are sampled randomly from the population; EvoOnly, where the model parameters kept fixed and only the evolutionary pipeline is applied; and TrainOnly, which removes the evolutionary mechanism and in-context prompt- ing, reducing the framework to pure GRPO reinforcement learning. These variants allow us to
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3.0
TopScore TopDiv
1.7
EvoOnly
day a WM
TrainOnly
Max Reward
2.5
Random
1.6
Ours
EvoOsly
Reward
Reward
TrainOniy
1.5
2.0-
TepSoore Top
14-
Lan
Random Ours
1.3
Circle Packing
Bosion Housing
Prohlems
Epoch
Epoch
Max reward of varients
(b) Circle Packing
Boston Housing
Figure 4: Ablation analysis of framework components. (a): Maximum reward achieved by dif- ferent ablation variants. (b): Curve of epoch-wise maximum reward on the Circle Packing task, highlighting the critical role of balancing diversity and quality for stable optimization. (c): Curve of epoch-wise maximum reward on the Boston Housing task, showing the necessity of combining reinforcement learning with evolutionary guidance.
disentangle the relative importance of reward-driven selection, diversity maintenance, evolutionary population updates, and reinforcement learning in driving overall performance.
Figure 4a reports the maximum reward achieved under different ablation settings. Across both tasks, all variants perform worse than our full framework, confirming the necessity each component. We next analyze the results task by task.
For the Circle Packing problem, high-quality solutions rely on diverse initial starting points for optimization algorithms. As shown Figure eliminating diversity (TopScore) significantly duces reward, since the search quickly collapses into narrow solution modes. In contrast, Random and TopDiv maintain higher diversity, enabling the model to extend from a richer of initial states. However, focusing solely on diversity also leads to citaability-viiibbe in the large variance of Top- Div and Ranrem-whereas TopScore and our full method (Ours) remain relatively stable. This instability disrupts training and prevents the model from finding strong solutions in later epochs. Moreover, we conducted a detailed analysis on the performance gap between OvenEvolve and HE- LIX in Appendix B which demonstrate the effectiveness of explicitly combining diversity with embedding model in our framework. These results highlight that balancing diversity and solution quality is critical for solving such problems.
For the Boston Housing task, strong performance requires careful parameter tuning and complex feature engineering, which typically emerge from iteratively learning from past experience. As shown in Figure[4c] disabling either reinforcement learning or evolution severely limits performance. With EvoOnly, the model remains bounded by its initial capacity and fails break through training bottlenecks. Conversely, with TrainOnly, the model cannot effectively accumulate knowledge in context and collapses during training. These results demonstrate that both parameter updates and in-context evolutionary guidance are indispensable for helping the model accumulate expertise and progressively refine its solutions.
4.3.2 SCALING EXPERIMENTS
Here, we discuss the impact of base model size on task performance. We evaluate our framework on two representative tasks, Magnetic Torque Maximization and Inductor Design, using the DeepSeek- R1-Distill-Qwen model family with 1.5B, 7B, 14B, and 32B parameters. As shown in Figure [ for the magnetic torque task, the reward steadily increases with model size, indicating stronger reasoning ability and more effective exploration. For the inductor design task, we observe reward plateau around 9.6. However, the mean reward continues to grow as model size increases, suggesting that larger models generate more valid and higher-qualite candidates. These results demonstrate that our framework exhibits scaling property: as the underlying LLM grows, the system can push the boundaries of scientific discovery by enabling more efficient and higher-quality exploration.

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10
10
Max Reward
Max Reward
Mean Reward
Mcan Reward
Reward
Reward
5
0
1.5
7
14
32
1.5
7
14
32
Model Size (GB)
Model Size (GB)
(a) Magnetic torque maximization
(b) Inductor design
Figure 5: Scaling analysis of model parameter scale on (a): magnetic torque maximization and (b):
inductor design tasks.
CONCLUSION
In this work, proposed HELIX, hierarchical evolutionary reinforcement learning framework with in-context experiences. By integrating reinforcement learning, evolutionary selection, and in- context trial incorporation, HELIX effectively balances exploration and exploitation, enables task- specific adaptation, and iteratively refines solutions. Extensive experiments across 20 tasks in five diverse categories demonstrate that HELIX consistently outperforms strong task-specific baselines and advanced proprietary models. Overall, HELIX shows strong potential for advancing open- ended scientific discovery by enabling iterative, diversity-aware exploration. Looking ahead, could provide foundation for broader applications in engineering, optimization, and autonomous research systems.
ETHICS STATEMENT
This work focuses on developing hybrid reinforcement learning and evolutionary framework for solving complex scientific problems. It does not involve human subjects, sensitive personal data, or proprietary datasets, and thus raises no direct ethical or privacy concerns. All datasets used are publicly available and widely adopted in prior research. All authors have reviewed and agree to abide by the ICLR Code of Ethics as linked above, and affirm that this submission complies with the principles of honesty, transparency, fairness, and responsible conduct.
REPRODUCIBILITY STATEMENT
Detailed descriptions of the experimental setup. task definitions, and evaluation metrics are provided in Appendix A and Appendix B
Source code will be available at atttppp://gttbbb...oo////eexxxxe
ACKNOWLEDGEMENT
This work was supported by Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (No. JYB2025XDXMI01, NSF of China Projects (Nos. 6255000425366003, 92370124, 92248303); Beijing Natural Science Foundation L247011; the High Performance Computing Center, Tsinghua University. J.Z was also supported by the XPlorer Prize.
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Angelo
Ziletti, Chris, Maggie, and
Will Cukierski.
Nomad2018 predict-
ing
transparent conductors.
attps://kaggle www/ cotp//tttmtttiiss///////
nomad2018 -orendiic--ppdiit------ ppwpdite tranuttt-taaaaareetttt cetnnppprret-codduutttrs 2017. Kaggle.
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A TRAINING AND EVALUATION DETAILS
Training. We primarily use DeepSeek-Distie Distilll-sstn---e--- and 32B as the backbone models in our experiments. The models are fine-tuned with the VERL framework (Sheng al. 2024 under the GRPO algorithm. Each model is trained for 80 epochs with fixed learning rate of X 10-6, updating all parameters. We set the KL coefficient in GRPO x 10 and the number of rollouts to 16. The rollouts are generated via VLLM (Kwon et al. 2023) backend with temperature equals 1.0 and top p equals to 0.95. Training was conducted using eight A100 GPUs for 14B models and sixteen H H100 GPUs for 32B models. For training efficiency, we use Pytorch FSDP (Zhao et al., 2023 with parameter offload and optimizer offload. Gradient checkpoint and Flash-Attention (Dao) used by default.
2024
Evaluation. The evaluation is performed on a Slurm Workload Manager system. For each job, allocate 4 Intel(R) Xeon(R) Platinum 8168 CPUs for execution and impose time limits for each task: five minutes for physics simulation, two minutes for machine learning and function minimiza- tion, and one minute for circle packing and symbolic regression. The execution time includes the time for tase-dependent evaluators to calculate reward. For the detailed evaluate metric and reward calculation, please refer to Appendix
B DEFINITION AND EVALUATION OF PROBLEMS
In this section we explain the detailed problem definition and evaluation metrics of all the tasks used in the experiment.
B. MACHINE LEARNING
We selected 3 classic machine learning datasets, and model has to write Python code to max- imize the F1 score for classification tasks and minimize the rooted mean square error (RMSE) for regression tasks. The details are described below.
B.1.1 ADULT INCOME
The Adult income dataset (Becker & Kohavi 1996) a well-known binary classification task. The goal is to predict whether a person's income exceeds $50,000 per year based various demographic features such as age, education, marital status, and occupation. The dataset is sourced from the 1994 U.S. Census and contains both categorical and numerical features, with some missing values.
The dataset itself contains a separate train and test split, We then load the train set for model's training and evaluate its result the test set. The reward is the Macro Fl score, defined as:
1 C 2 P. Re
R R= C= Pat Re
where Pas Rs are the precision and recall for class and C' = is the total number of classes.
B.1.2 BANK MARKETING
The Bank marketing dataset (Moro et al. 2014) is another binary classification problem. It includes data from a Portuguese bank's direct marketing campaigns, where the objective predict whether a client will subscribe to term deposit. This dataset is characterized by high number of categor- ical features and a significant class imbalance, making it good benchmark for evaluating model performance under challenging real-world conditions.
To ensure robust evaluation, we use a 5-fold cross-validation strategy with StratifiedKFold in sklearn to handle the class imbalance. The data is randomly split into five folds, maintaining the same class distribution each fold as in the original dataset. The model trained and evaluated five times, with each fold serving as the test set once. The final reward is the average of the Macro Fl scores obtained from all five folds. a task fails to produce a result in any fold, its reward is considered be 0 for that fold. The final result is the Macro F1 score, as defined in equation
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B.1.3 BOSTON HOUSING
The Boston housing dataset (Harrison Jr Rubinfeld 1978) is a classic regression problem. The lask is to predict the median value of owner-occupied homes in Boston suburbs, based on 13 features. These features include per capita crime rate, number of rooms per dwelling, and the proportion of non-retail business acres. While the original dataset is no longer widely used for research due to ethical concerns, it remains a common benchmark for teaching and evaluating regression models.
To evaluate model performance, we use 5-fold cross- validatoio strategy with KFold, splitting the data into five folds. The model is trained and evaluated five times, with each fold serving as the test set once. The final reward for this task is the average of the scores from all five folds. The reward is calculated using the following formula:
R = - 1o (RMSE + 10 10-11),
(8)
where:
RMSE
yy 01)²
This reward metric is designed to penalize larger RMSE values while rewarding smaller ones. If task fails in any fold, its reward is considered be for that fold.
B.1.4 TRANSPARENT CONDUCTORS
The Transparent Conductors dataset (Ziletti et al. 2017) is motivated by the need for accelerated discovery materials that simultaneously exhibit optical transparency and electrical conductiv- ity-two properties that are typically at odds. Such materials are central to modern technologies including photovoltaic cells, LEDs, sensors, touch screens, and display panels. Despite their im- portance, only limited number of compounds are currently known to meet the desired trans- pardncy-conductivity trade-off, making data-driven exploration an appealing alternative to costly experimental quantum mechanical searches.
The dataset contains computationally derived information for 3,000 candidate materials belong- ing to the sesquioxide alloy family [Al
where the compositional ratios satisfy x + y + 2 = and the total number of atoms in the unit cell ranges from to 100. These materials of particular interest due their large bandgaps, chemical stability, and relatively low produc- tion cost. Each entry includes crystallographic descriptors (e.g., space group, lattice parameters), compositional ratios, and structural characteristics, offering rich feature space for modeling.
The task is to predict two key target properties for each material: (1) formation energy, which reflects thermodynamic stability, and (2) bandgap energy, which determines visible-range transparency. Ac- curate prediction of these quantities enables efficient screening of new transparent conductor candi- dates without the need for expensive
theory (DFT) calculations.
Model performance is evaluated using the root mean squared logarithmic error (RMSLE), computed column-wise for the two target properties, For single target, the RMSLE is defined as:
RMSLE
(log(p: + +1 log(a; + 1))²
(10)
2=1
where is the number of samples, Ps denotes the predicted value, and a; the ground-truth value. The final reward for model training is:
R RMSLE
(11)
B.2 PHYSICS SIMULATION
To test the model's capacity for geometric reasoning and ability utilize physics prior knowledge to discover better designs, we proposed the following physics simulation tasks. These tasks mainly require the model to generate yaml representation of a complex geometry under certain constraints to maximize the reward. We utilize COMSOL Multiphysics@ (COMSOL AB 20224), a commercial FEA software for industrial multiphysics simulations, for the evaluation backend.
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B.2.1 ACOUSTIC DEMULTIPLEXER
This task aims design an acoustic demultiplexer. The demultipexer is data distributing device which takes acoustic energy from the input port and distributes different frequency bands to the specific output port. The model is asked to propose the cavity geometry within a circular domain as seen in Fig. [ to maximize the acoustic pressure at output port 2 while minimizing the pressure at output port The input acoustic pressure level is set at port 1, and the frequency level is set to 7500 Hz.
Port 2: Output
Pa
1
Port 1: Input
-1
Port 3: Output
Figure 6: The RMS pressure field of an acoustic demultiplexer at frequency level 7500 Hz. The RMS pressure field in log scale is proportional the acoustic power.
The model is guided by the following reward R where P; is the power output at port i, Prms is the Root Mean Square (RMS) pressure field, p is fluid density, and is sound speed.
Prms dl
P=
port pc
(12)
log (P2) logio (Pa)
0.292
We use value 0.292 on the denominator of Eq. 12 to normalize the reward. And Fig. shows symmetric design with 7 circular cavities in the computation domain, producing equal acoustic pressure the two output ports and thus R O. Notice that LLM is not limited by the circular cavity pattern, and is prompted to freely explore any viable cavity geometries within the computation domain.
B.2.2 MAGNETIC TORQUE
This task aims to design the geometry of an iron core that generates large torque when subjected to a uniform magnetic field. Fig. 7 shows the problem setting, an example iron core geometry and the corresponding magnetic flux density norm field. uniform magnetic field intensity of H [0, 1e5] A/m is applied to the circular boundary. The iron possesses large permeability > No distorts the magnetic flux density field within the circular air domain. The distorted B thus applies torque the iron core, which be obtained from Comsol by solving the static Maxwell's equations.
To guide the model reinforcement learning and evolutionary search, the following reward R is com- puted as below where T Maxwell stress tensor, r position vector, and represents magnetic torque which is simplified to in 2D simulations:
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Magnetic flux density norm
HL= 0 A/m H, 1e5 A/m
Air
Figure 7: The magnetic flux density field generated by an iron core subject to uniform magnetic field boundary condition. The distorted magnetic flux density field then applies torque on the iron core.
1
T
(BB
B²
Mu
2
(13)
9241.99 A f r x (T -i)dA]
9241.99 A
We use a value of 9241.99 on the denominator of Eq. 13 to normalize the reward. Notice that a perfectly symmetric iron core (for instance circle) would have 0. Therefore, expect to train and evolve the LLM to produce a highly irregular iron core geometry to generate large magnetic torque values. We minimum area of 2e m² avoid naive designs.
B.2.3 BEAM BENDING
This task aims to design the cross section geometry of a cantilever beam subject to a superposed loading of bending moments M, and My. shear forces T₃ and T, along the in-plane directions, and twisting moment T32 along the out-of-plane direction. The cantilever beam is assumed to be
linear elastic with Young's modulus GPa and Poisson's ratio 0.3. Fig. X shows example beam cross section design and the von Mises stress distribution as calculated from Eq. [4] solved using the Beam Cross Section module Comsol. As the cross section stays in the x-y plane, Taz- yyu and Tzy take O values.
1 Oem = Vo [o - owy)² + (ouz )2 (awy ozz) (T + 22 + T32
(14)
18.2
The reward is set to be R moment of inertia, I2 is the smallest second moment of inertia. We use value of 1.32e-3 on the denominator to normalize the reward. I, and /2 represent the beam's 's largest and smallest resistance over different bending loading directions, and can be calculated from the stress field following the classical beam bending theory (Bauchau & Craig 2009). We set a minimum area of 2e m² to avoid naive designs.
1 where A is the cross section area, J is the largest second
B.2.4 PERIODIC HEAT
This task aims to design the unit cell geometry of a periodic meta-material for best effective thermal conductivity. The base material is assumed to be aluminum with density 2700 kg/m² and thermal conductivity 238 W/mK. Fig. X shows an example 2D unit cell geometry which will extruded in the z direction to form the 3D unit cell. The resultant temperature distribution and effective
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0.3
Beam cross section shape
Mises stress
Figure 8: The Mises stress field generated by applying bending moment, shear force, and twist-
ing moment on cantilever beam cross section design.
Temp (K)
294
Design
Load 1
293
Load
Load 3
Figure 9: Temperature distribution of the meta-material under three loading conditions. The effec- tive properties are calculated based temperature distributions according the homogenization theory.
properties are solved using Comsol based on the homogenization theory. The results are calculated from a temperature difference boundary conditions along x, y, and z directions.
trace(kefs)
R 0.178 Peff
(15)
where keff is the homogenized effective thermal conductivity matrix, and Peff is the effective density, which simply equals to the percentage of volume filled by aluminum. We use a value of 0.178 on the denominator to normalize the reward. This objective function targets to maximize the thermal conductivity along x, and z directions under limited material usage. We set maximum effective density Peff 2000 kg/m² to avoid naive designs.
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B.2.5 INDUCTOR
This task aims to design an inductor which is a critical component in power electronics. Fig. T0 shows example inductor consisting of iron and coil windings in cylindrical coordinate. A sinusoidal current excitation is supplied to the coils at a frequency of 1000 Hz and magnitude 500 A. The iron core possesses a nonlinear magnetization curve with an initial permeability of 663 H/m and saturates at 5 T. The resultant magnetic field is calculated using Comsol by solving the Maxwell's equations in frequency domain. The model is asked to propose the optimal iron core geometry well the placement of the coil windings (coil shapes are fixed) to produce the maximum inductance with limited material usage.
T
1.8
Syn Commenss axis
Iron core
Coil
Magnetic flux density norm
Figure 10: The magnetic flux density norm field generated by inductor. The copper coils excited by a 500 A, 1000 Hz sinusoidal current.
L 43.11 V
0.5 (B, H B Ho B. H.)dV
(16)
43.11-
The reward calculation is shown in Eq. 16 where B,, Bg, and B, cylindrical components of magnetic flux density field, and H,, Ho. HT are components of magnetic intensity field. Both fields value of 43.11 the denominator
take complex values for frequency domain response. We to normalize the reward. The numerator stands for the inductance which is a volume integral of magnetic energy. We set minimum iron core volume le m² to avoid naive designs.
B.3 CIRCLE PACKING
The objective of these tasks is to pack a fixed number of circles specific domain and maximize the sum of the radii of these circles. The circles cannot overlap with each other exceed the domain boundary. All the centers and radii can change as long as the constraints are satisfied.
Formally, let 26 be the number of circles, (sa Jis be the coordinates of centers and (rahien be the radii. The objective can be written as:
R
Tis
(17)
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while the constraint is
V (xt - x2)² (y: - ri try <ii<jsn
21 - ri 2 0,
vI n VI V1 SISiSn VI n.
(18)
+ ri 1,
- ri 2 00 V/ + r; 1,
for the packing in unit square, and
(z: x2)2 + (yo - v,) 2ni
A1 viii<s
(19)
Vaî + of +ms1.
viiisn
for the packing in unit disk.
B.4 FUNCTION MINIMIZATION
These tasks require the model to find an effective algorithm to locate the global minimum of complex function with various local minima. For given function f(x") and the model's prediction x", The evaluation metric defined as:
[x(x)] [(xx) [5(**) f(x*)|
R
(20)
This metric suitable for distinct functions with varying scales [x(x))], satisfies 0 s R and if the model successfully finds the global minimum, the reward will be R 1.0.
B.4.1 EGGHOLDER FUNCTION
The Eggholder function is a classical task for evaluating evolutionary optimization algorithms with various local minima. It can be defined as:
f(x) (xx2 47) sin( (x2 47) 21 2ı sin( V (22 47))),
(21)
with constraint -512 < 21 I 512 and global minimum f((512, 404.2319) 959.6447
under such constraint.
Figure iillustrate the landscape and the global minimum point of the Eggholder function.
B.4.2 MISHRA'S BIRD FUNCTION
The Mishra's Bird function is classic test function used in optimization to evaluate the performance of algorithms. known for having a unique "bird-shaped" landscape with multiple local minima and single global minimum. It's often used to test algorithm's 's ability avoid getting stuck in suboptimal solutions.
The function is defined as:
f(x) (in( cos(ss)e(t-snn(e))) (x1 x22)
(22)
with the constraints:
-10 < 21 0
-6.5 < x2 <
(23)
x + 25.
The global minimum -((-3.1302-115522) ~ 106.76645
Figure shows the landscape of the Mishra's Bird function and its global minimum point.
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1000
Global Minimum
750
1500
500
1000
250
500
Function Value
0
250
-500
-500
-100
-750
1500
-1000-
2000
1000
-500
0
500
1000
Figure 11: The landscape and global minimum point of Eggholder function with constraints -512 <
x, < 512
B.4.3 KEANES BUMP FUNCTION
The Keanes Bump function is challenging, non-convex test function commonly used to evaluate the performance of optimization algorithms handling high- high-dimensional problems with complex constraints. The function's 's landscape is highly irregular, containing numerous local minima, and its feasible region is a small, irregular subset of the search space.
Let be the dimension of variables and f Rd - I. the function is defined as:
Dd iml cos' 1 (x,) 211 cos? (x;))
f(x)
(24)
Σラ ix?
with the following constraints:
x4 S 10, visisd
0; S 7.5d
(25)
d
II 0.75.
The global minimum is located within the feasible region, which is small, bounded area defined
by these constraints. The image in this document, Figure I3 shows a two-dimensional visualization of the function's 's landscape. However, for our experiments, tested the function in its 10-D, 20- D. and 30-D versions, where the complexity increases significantly. The global minima and their corresponding function values for these dimensions listed below.
10-D Version: The global minimum value is approximately -0.47731032..
20-D Version: The global minimum value approximately -0.803619004.
30-D Version: The global minimum value approximately -0.81 8056222.
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0
Global Minimum
75
30
50
-2-
25
25
Function Value
4
0
-25
6
-50
-8-
50
75
25
100
10
-10
8
6
-2
0
Figure 12: The landscape and global minimum point of Mishra's Bird funcction with constraints
I1 [-10, 0). x2 E [-6.5, 0j and of + 2 2 25.
B.5 SYMBOLIC REGRESSION
In this task, the model has to uncover symbolic mathematical expressions from observational data. The benchmark and baselines are provided by (Shojaee et al. (2025), which includes equations and data in chemistry, biology, physics and material science domains. In each category, several cases created, each containing its own train and test sets generated by the same underlying equation. The model trained on the train set has to propose an expression minimize the normalized mean square error (NMSE) on the test set, which is defined as:
ΣN (ir y:)
NMSE
(26)
Σ (34 7)
where is the number of observations in the test set.
To ensure a fair and robust comparison with the benchmark paper's results, we use the median of the NMSE calculated across all tasks within the same category e:
NMSE median (MSEE,,1 NMSE NMSE,,,)
(27)
The reward we used for reinforcement learning for category c then set to:
R. logau (NMSE.).
(28)
In the benchmark, all the methods have a limit of 1000 trials for each single case, and obey the same rule in our experiments, adjusting the number of training steps accordingly.
C DESCRIPTION OF TASK SPECIFIC BASELINES
In this section, we introduce the task-specific baseline methods and describe their implementation details.
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10
0.00
Global Minimum
-0.05
8
-0.10
Function Value
-0.15
-0.2
4
0.25
2
-0.30
00.35
2
4
8
10
Figure 13:
The landscape and global minimum point the 2-D Keanes Bump function. The feasible region, a small part of the search space, is the only area with finite function values.
Machine Learning. For machine learning benchmarks, we evaluate two interpretable yet com- petitive models: LightGBM (Ke et al. 2017). gradient boosting framework widely adopted in practice, and Rulc-based Representation Learner (RRL) (Wang et al.) (2021), which learns discrete non-fuzzy rules via gradient grafting achieve both scalability and interpretability.
Physies Simulation. For ppysiccs-relttd optimization problems, ules in COMSOL Multiphysics: parameter search and topology optimization. For parameter search, first parameterize the geometry based initial solutions provided by human experts, and then optimize within the search space defined by these parameters. For topology optimization, human experts specify deformable geometric regions, while COMSOL applies its built-in topology opti- mization solvers to iteratively refine the structure.
two widely adopted mod-
Circle Packing. We consider two strong baselines: Sequential Least Squares Programming (SLSQP) (Lawson & Hanson| 1995) and Genetic Algorithm (GA). SLSQP formulates circle pack- ing as a constrained optimization problem, maximizing the sum of radii subject to boundary and non-overlap constraints. The GA baseline encodes circle positions and radii, evolves feasible population with selection, crossover, and mutation, and evaluates fitness by the total radii.
Function Minimization. We adopt two standard constrained optimization solvers from scipy - opttimmmmmm... ze: Sequential Least Squares Programming (SLSQP) and the trust-constr method (Conn et al. 2000). Both are widely used gradient- -based methods that provide strong task-specific baselines for function minimization.
Symbolic Regression. For symbolic regression tasks, directly results reported in LLM- SRBench (Shojace et al., 2025), obtained by GPT-4o-mini running two recent methods: LaSR (Grayeli et al. 2024), which enhances evolutionary search with LLM--gidded concept discovery, and
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LLM-SR (Shojace al. 2024 which combines LLM scientific priors with evolutionary equation search. These represent competitive state-of-the-ar baselines for symbolic regression.
D EXAMPLE OF PROMPTS USED IN EACH EXPERIMENT
In this section, we will demonstrate the prompts we used in our experiments.
D.1 MACHINE LEARNING
Prompt for task Adult Income
You are an expert software developer tasked with iteratively improving + a codebase.
Your job is to analyze the current program and suggest improvements based feedback from previous attempts. Focus on making targeted changes that will increase the program's + performance metrics.
Respond in the following format: <think>
</think>
<answer>
</answer>.
Problem Description
You are an expert traditional machine learning.
Your task is to build predictive model using the **Adult Income
Dataset** (also known the "Census Income" dataset).
This dataset contains demographic and employment-related attributes
collected from the 1994 U.S. Census database.
The goad is to **predict whether client will subscribe to term deposit** ("y" col um:: yes/no) based on demographic and markecing-relltte features.
The goal is to **predict whether person's income exceeds \$50K per
year** (income column: >50K / <=50K) based on individual and employment features.
## Dataset Features
### Target variable:
**income**: ">5CK", '<-50K". (Parsed to 1/0 in Program)
### Input variables:
1. **age** + (numeric)
Age of the individual.
*workclass*** (categorical)*
Type of employment:
'Private', "Self-emp-not-incy; 'Self-emp-ine", 'Federal-gov", "Local-gov",
"State-gov", 'Without-pay', 'Never-worked',
3. **fnlwg*** + (numeric) Final sampling weight indicates the number of people represented by this record.
**education** * (categorical) kducation level:
"Bachelors', "Some-college', '1lth", 'HS-grad", "Prof-school", "Assoc-acdm", "Assoc-voc', "9th", "7th-8th", "12th", "Masters", '1st-4th', '10th", 'Doctorate', '5th-6th', "Preschool',
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meduccttion-nue* (numeric) Encodes years of education.
6. +marital-statusrs * (categorical)*
Mazried-civ--pousee "Divorced", "Kever-married', "Separated", "Wldowed,, "aariss-ouuc-benn', "Marridd-A-spouus"
7. *occupation** (categorriaa))
TTehh "Craft-repair", 'Other-service', 'Sales',
"xxec-managerial",
proof-spccialty", "Handlers-cleaners) "Mahhnn--ppinsppct"
+ 'Adncllerical,,
"Farming-fishing' coanssport-oviingg 'Priv-house-serv', "Protectivesserv'
Armed-Porces".
8. #relationshipet (categgrical)) "ife", "Own-child", "Husband", 'Not-in-family "Other-relative", UUmmarried'.
9. **rac*** (categorical)> "White', "Asian-Pac-sslndee Amer--nndannEskinng "Other", "Black".
10. ***ex** (categorical) "emale", "Male".
11. -capital-gainn* (numeric)
Income from capital gains.
12. **capital-loss** (numeric) Losses from capital assets.
13. whours-per-week** (numeric) Number of working hours per week.
14. -nativv--couttzys (categorical)
Waited-States', 'Cambodia', 'England", "Puerto-Rico', "Canada", + "Germany",
"Outlying-US (Guam-USV1-etc)) "Idia' 'Japan', "Greece', 'South', "China", "Cuba", "Iran', "Honduras", "Philippines', 'Italy', "Poland', + "Jamaica",
"Vietnam, "Mexico", 'Portugal', 'Ireland', "France", "Dominicar-Republicy "Laos", "Ecuador", "Taiwan', 'Eaiti' 'Columbia', "Hungary", "Guatemala', "Nicaragua', "Scotland", 'Thailand', "Yugoslavia", 'El-Salvador', "Trinadada Tobagoo "Peru', "Hong', "Holand-kecherrladds.
## Additional Notes
You may add, delete, or modiřy functions arbitrarily, but the program must still contain the suu__modellf function.
If you want to use new packages, please import then explicitly. Try different **data preprocessing+*, **feature engineerings*, and **modeling techniques** to improve performance,
Pay attention. to Missing values: represented as "?", Handle them properly.
Pay actention to Categorical encoding: many features categorical; choose effective encoding strategy.
Task
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Write machine learning model to **predict** whether a person's + income exceeds \$50K.
You will be giver. **starter program** in Python. Your goal is to **improve this program** maximize the **F1-score** on the test set.
Your code execution time should **not exceed 60 seconds**. You MUST the exact SEARCH/REP REPLLACE fiff format shown below when modifying code:
<<<<<<< SEARCE
Original code to find and replace (must match exactly)
replacement code
>>>>>>> REPLACE
Current Program
Status: (currert_status]
"'python
jcurrent_
Prompt for task Bank Marketing
You expert software developer tasked with iteratively improving
+ a codebase.
Your job is to analyze the current program and suggest improvements
+ based on feedback from previous attempts.
Focus making targeted changes that will increase the program's performance metrics.
Respond in the following format: <think>
</think>
<answer>
</answer>.
Problem Description
You are an expert traditional machine learning.
Your task to build predictive model using the **Bank Marketing
Dataset**.
This dataset contains information collected from direct marketing
campaigns conducted by Portuguese banking institution.
The goal is to *spredict whether client will subscribe deposit** ("y" column: yes/no) based on demographic and marketing-relates features.
Dataset Features
Input variables:
### bank client data:
age (numeric)
job type of job (categorical: "acmin.", "blue-collar", "entreprereur", "housemaid", "management", "rstired", "aelf-employes oyed", "services", "student", "technician", "unemployed", "unknown")
marital marital status (categorical: "divorced", "married",
"single", "unknown"; note: "divorcec" divorced widowed)
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education (categorical: "basic.4y", "basic, 6y", "basic. 9y", "high.school", "illiterate", "professional. course", sewiverrityydegrree "unknown")
default: has credit in default? (categorical: "yes", "unknown") "unn""""oo""""""""
housing: has housing loan? (categorical: "no", "yes", "unknown")
loan: personal loan? (categorical: "no", "yes", "unknown")
### related with the last contact of the current campaign:
contact: contact communication type (categorical:
"cellular", "telephone")
month: 1ast contact month year (categorical: "jan", "feb", "nov", "dec")
"mar",
10 day_of week: last contact day of the week (categorical:
"mon","tue"" "tue", "wed", "etuu",frii"
duration: last contact duration, in seconds (numeric). ### other attributes:
12 campaign: number of contacts performed during this campaign and
4 for this client (numeric, includes last contact)
13 - pdays: number of days that passed by after the client was last contacted from previous campaign (numeric; 999 4 was not previously contacted)
client
14 previous: number of contacts performed before this campaign and for this client (numeric)
the previous marketing campaign 4 (categorical: "fai lure", "nonexistent", "success")
pouteomm: outcome
### social and economic context attributes
16 ump.var.rate. employment variation rate - quarterly indicator (numeric)
cons.price. iix: consumer price index monthly indicator
4
(numeric)
18 cons. conf. idx: consumer confidence index monthly indicator (numeric)
19
curibor3m: curibor month rate daily indicator (numeric)
nr cr.moyoy number of employees quarterly indicator
4 (numeric)
## Task
Write machine learning model to *xpredict** whether a client subscribes to term deposit.
You will be given **starter program** in Python. Your goal
**improve this program** maximize the **F1-score** = on the test set.
Your code execution time should **not exceed 60 seconds**. You MUST the exact SEARCH/REFLACE diff format shown below when modifying code:
<<<<<<< SEARCH
Original code to find and replace (must match exactly)
New replacement code
>>>>>>> REPLACE
## Additional Notes
You may add, delete, or modify functions arbitrarily, but the + program must still contain the 'run_model ()* function.
If you want to use new packages, please import them explicitly.
Tzy different **data preprocessing** **feature engineeringss, and **modeling techniques** improve performance.
Current Program
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Status: (current_status) "python ;cuzrent_program)
Prompt for task Boston Housing
You are an expert software developer tasked with iteratively improving a codebase.
Your job is to analyze the current program and suggest improvements
based feedback from previous attempts. Focus on making targeted changes that will increase the program's performance metrics.
Respond in the following format: <think>
</think>
<anawer>
</answer>.
Problem Description
You are an **xppert in traditional machine learning**.
Your task is to build *xpredictive regression model** using the **oston Housing Dataset**.
The **Boston Housing Dataset** contains information collected by the
U.S. Census Service concerning housing in the Boston, Massachusetts area.
The goal is to **predict the median value of owner-occupied homes**
("MEDV' measured in ($1000s) based on various demographic, economic, and geographic factors.
## Dataset Features
The dataset contains **13 numerical and categorical features**. Some of them may have missing values (nan in dataframe)
1. **CRIM* Fer caplta crime rate by
**ZN** Proportion of residential land zoned for lots over 25,000 sq.ft.
+
3. **INDUS** { Proportion of non-retail business acres per town
4. **CHAS** Charles River dumny variable (1 if tract bounds river; 0
otherwise)
**NOX** Nitric oxides concentration (parts per million)
6. **MM** Average number of rooms per dwelling
7. **AGE** Proportion of owner-occupied units built prior to 1940
8. **DIS** weighted distances to five Bostoo employment centres
**RAD** Index of accessibility
radial highways
10. **TAX** Full-value property tax rate per \$10, 000
11. **PTRATIO** | Pupil-teacher ratio by town
12. **LSTAT** Percentage of lower status population
13. **MEDV** **Target variable**: Median value of owner-occupied
homes \91CCCs
## Task
You will provided with **starter Python program**.
Your objective is to **improve the program** to build a more accurate **regression model** for predicting "MEDV". Your improvements should focus on **maximizing the RMSE score** the **test set** (RMSE
2 logl0 (RMSE))
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## Requirements
Your code execution time **must not exceed 60 second***.
You MUST use the **SEARCH/BEPIACE diff format** exactly as shown below when modifying the code:
<<<<<<< SEARCH
$ Original code to find and replace (must match exactly)
===_===
replacement code
>>>>>>> REPLACE
## Additional Notes
You **may add, delete, or modify functions** as needed, but the progran **must still contain** the rrun_model ()" function. If you want to use new packages, please import them explicitly.
Usable packages: pandas, numpy, sklearn, scipy, statsmodels, xgboost, lightgbm, catboost, category sttegorr_enncders, imbalanced-learn Tzy different **data preprocessinge*, **feature engineerings+, and **modeling techniques** to improve performance.
## Current Program
Status: (current_status)
"pyyhoo Baazzeet_prograate
prompt for task Predict Transparent Conductors
You are expert software developer tasked with iteratively improving codebase.
Your job is to analyze the current program and suggest improvements
+ based on feedback from previous attempts.
Focus making targeted changes that will increase the program's performance metrics.
Respond in the following format: <think>
</think>
<answer>
</answer>.
# Overview
Description
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Innovative materials design is needed to tackle some of the most important health, environmental, energy, social, and economic challenges of this century. In particular, improving the properties of materials that are intrinsically connected to the generation and utilization of energy is crucial if we are to growing global demand.
mitigate envienttntal damage due Transparent conductors are an important class of compounds that are both electrically conductive and have a low absorption in the visible range, which are typically competing properties. A combination of both of these characteristics is key for the operation
variety of technological devices such photovoltaic cells, light-emitting diodes for flat-panel displays,
transistors, sensors, touch screens, and lasers. However, only
small number of compounds are currently known to display both transparency conductivity suitable enough to be used transparent conducting materials.
Aluminum (Al), gallium (Ga), indium (In) sesquioxides are some of the
most promising transparent conductors because of combination of both large bandgap energies, which leads optical transparency over the visible range, and high conductivities. These materials are also chemically stable and relatively inexpensive to produce. Alloying of these binary compounds in ternary or quaternary specific
mixtures could enable the design of new material composition with improved properties over what current
possible, These alloys are described by the formula (A1_x Ga_y where x, Y, and z can vary but are limited by the constraint xtytz = 1. The total number of atoms in the unit cell, / (where is an integer), is typically between and 100. However, the main limitation in the design of compounds is that identificatio and discovery of novel materials for targeted applications requires an examination of enormous compositional and configurational degrees of freedom (i.c., many combinations of x, y, and 2). To avoid costly and inefficient Arial-and-erron of synthetic routes, computational data-driven methods can be used to guide the discovery of potentially more efficient materiais to aid in the deveiopment of advanced (or totally technologies. In computational material science, the standard teol for computing these properties is the quantum-mechanice method known as density-Eunetional theory (DFT). However, DFT calculations are expensive, requiring hundreds thousands of CPU hours supercomputers for large systems, which prohibits the modeling of a sizable number possible compositions and configurations. As a result, potential (AAl_ Ga_y In_z)_112N) 10_ [[_((3 materials remain relatively unexplored, Data-driver: models offer alternative approach to efficiently search for new possible compounds in targeted applications but significartly reduced computational cost.
This competition aims to accomplish this geal by asking participants
to develop apply data analytics/data nining/machine-leamee
models for the prediction of two target properties: the formati a new material)
energy (which an indication of the stability
and the bandgap energy (which is an incication of the potential
for transparency over the visible range) to facilitate the discovery of transparent conductors and allow for advancements in the above-mentioned technologies.
## Evaluation
Submissions evaluated the column- wiie reot squared Logarithmic error.
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The RMSLE for single column calculated
cNoqrc([(frac(1]] (oo 1))) \log( ( (a_i+1] 2
where:
snş is the total number of observations
sp_i$ is your prediction
$a_i$ is the actual value
S'oo (x) the natural logarithm of Sx$
The final score is the mean of the RMSLE over all columns (in this case, 2).
Dataset Description
High-quality data are provided for 000 materials that show promise as transparent conductors. The following information has been included:
spacegroup: Crystallographic space group number describing the symmetry of the material.
Total number of atoms in the unit cell. percent_atco_al, percent_aton_ga, percent_atom Relative composition of A1, Ga, and In in the material (remaining fraction is 0)
rumber_of_total_e
serriceeeeectr... 1 ang, lattice lattice_vector vector 2 _ang, lattice lattice_vector 3 3_ang: Lengths of the three lattice vectors (in angstroms), describing the unit cell dimensions. puctico_aggle_btta_dee
Angles between the lattice vectors (in dogrecs), defining the unit cell geometry.
A domain expert will understand the physical meaning of the above information but those with a data mining background may simply use the data input for their models.
The task for this competition
predict two target properties:
Formation energy (an important indicator of the stability of material)
Bandgap energy (an important property optoelectronic applications)
Task
## Additional Notes
You **may add, delete, or modify functions** as needed, but the program *mmust still contain** the rrun_modell ()* function. packages, please import them explicitly. Usable packages: pandas, numpy, sklearn, scipy, statsmodels, xgboost, lightgbm, catboost, category category__enooeeee encoders, imbalanced-Leacn Tzy different **data preprocessing** **feature engineering**, and *modeling techniques** to improve performance.
If you want
## Requirements
Your code execution time **must not exceed 60 seconds**.
You MUST use the **SEARCH/EEPIACE diff format** exactly as shown below when modifying the code:
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<<<<<<<
Original code to find and replace (must match exactly)
+ New replacement code >>>>>>> REPLACE
## Current Program
Status: (currert_status) "'python jcurrent
D.2 PHYSICS SIMULATION
Prompt for task Inductor
You are helpful Al Assistant that provides wel all-reeaooed and detailed responses.
You first think about the reasoning process as an internal monologue + and then provide the user with the answer. Respond in the following format: <think>
...
</think>
<anawer>
</anawer>.
## Task Description
You are a helpful AI Assistant and scientist with strong physical background and wonderful geometric desi ideas.
You are asked to generate the geometry design of component using
yaml files under certain constraints. You will first create geometries of your design, and then assign functions to the geometries according to the specific requirements.
Your final answer should contain yaml file enclosed in ""yaml An (your code) ... The yaml file should have at least two parts: geometry and selection. The specific requirements are as follow:
geometry: 1ist of objects with type and type-speci fic parameters. The types and parameters are as follows:
Polygon: (2D) You can use it to create rectangles, triangles, etc. table: Ordered list of vertices as 1x, yl points. The polygon is formed by *xconnecting consecutive points** (p_i->p_(1+1) and **automatically closing** the shape (p_n->p_1).
fillet: (Optional) A list of [i, r] tuples, where i is the index (starting from 1) of a polygon vertex defined in the above table, and is the fillet radius for that corresponding vertex.
Ellipse: (2D) You can use it to create circles. semiaxes: (horizontal, vertical] axis lengths
pos: [center_x, center_yl center position
rot: (Optional) Rotation angle (degree) counterslockwis
angle: (Optional Angular span (degree) counterclockwi se. e.g. by setting angle-180 you can draw a upward semici rcle. LineSegment: (1D)
coordl: [start start_yl
coord2: [end x, end
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CircularAra: (1D)
r: Radius
anglel: start angle (degree) counterclockvise, 0 degree
+ represent positive direction of X-axis.
angle2: End angle (degree) counterclockvise
CubicRezier: (1D)
p: Control points as x2,,, [y0.y1,y2,y31] w: Weight values as w2,
Interpolationcurve: (1D)
table: Ordered 1ist of [x,y] points interpolate through.
The curve will pass every points smoothly (polynomial
4 interpolation for x and y).
Parametrice (1D)
parname: Name of paraneter
parmin: Minimum value of parameter
parmax: Maximum value of parameter
coord: Expressions about the parameter like ("expression_x", "expression_y"). Trigonometric functions here use radians ConvertToSolid: (2D) Geometry formed by end-to-end connected ID curves.
geometries: A dictionary
1D geometries (using the same structure as the top-level geometry section, recursive). +*They Must connect end-to-end and form simply connected
-
space**.
Union: (2D) Union
geometries.
geometries: A dictionary of geometries (recursive).
Intersection: (2D) Intersection of 20 geometries.
geometries: A dictionary of geometries (recursive).
Difference: (2D) Difference of the 2D gecmetries.
geometries add:
add: A diczionary of gecmetries to keep (recursive).
gcometrles_aubrraet A dictionary of geometries to subtract (recursive).
After **geometry** was created, the shapes will be splitted into *enon-overlapping connected regions**.
Overlapping 2D shapes create new regions (e.g., two intersecting circles regions)
Enclosed 2D shapes spli: regions (e.g., circle inside polygen
+ 2 regions: circle interior + polygen-ring)
10 curves through 20 shapes create sub-regions (e.g., line segment through rectangle alternating regions)
The **regions** can be represented by the following ways:
point: You can select an interior point of the region to
represent it. The point should never on boundaries/corners. One point per region suffices.
geometry: The 2d shapes you created might be splitted into several regions, You can select the geometry to represent the regions in it.
2. selection: After regions functions regi ons using selections.
created, you will assign di fferent
UnionSelection: Union of all the regions selected below. points: (Optional) List of |x,yl points representing distinct regions.
geometries: (Optional) List of 2d geometry names you created above. By listing geometries here, you can select all the + region this geometry contains.
selections: (Optional) List of other selection names you created.
IntersectionSSelettion: Intersection all the regions selected below.
same parameters as UnionSelection
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Di DifferenceSelectionn Select the regions in Add but not in Subtract.
add: same parameters as UnionSelection.
subtract: same paraneters as UnionSelection.
Finally yaml file will like the following sample:
''yaan
geometry:
unil: Name of this geometry
type: Union
geometries: $ create geometries recursively below uni ell: $ Name of the first ellipse to union
type: Ellipse
Type of this geometry
semiaxes: 12.0, 1.01 Specific parameters
pos: [1.0, 1.0]
uni poll: / Name of the second polygon
type: Polygon
table:
[-1.0, -0.3]
[2.0, -1.0]
[1.0, 1.0]
linel: This line spiits the ellipse into 2 regions.
type: LineSegment
coordl: [1.0, 2.0]
coord2: [3.0, 1.0]
selection:
sell: Name of this selection
type: DifferenceSelection
add:
geometries:
unil Select all the regiens in unil
subtract:
points:
[2.5, 1.5] # Remove the region where (2.5, 1.5) in. This
region is part of ellipse but splitted by the line segment.
## Geometric Design of Inductor2d
You are asked design an inductor. The objective is to maximize the inductance value the inductor which is calculated as 50.5ereal (Brsconj (Hr) +conj (Bph))+Bphh *Bphi (Hz) *B2) /V$ with B, и to be the magnetic induction intensity and magnetic field intensity and V be the volume of the core.
The geometry should be designed inside a semicircle of radius 0.35m centered at (C, opening in the positive x-direction. Then we will generate 3D geometry by rotating the semicirsle around the axis x=0
You are required to give the geometry of the core, and the location of the coils. After create the geometry, you should select the regions of the core. **The Name of the selection must be 'core"
Finally, you need give the center the coils, which are circles with radius 0.01m. You don't 't need to give the geometry of the coils.
The constraints
follow:
1. There's no overlapping of different coils. There must be 12 coils in total.
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2. The core and the coils should not overlap adjacent.
3. The geonetry and coils should placed inside the semicircle of
+ radius 0.35m centered at (0,0), opening in the positive
x-direction.
4. The volume of the core should be more than 0.001 m3 This means that are **extremely thin extremely fine** not allowed.
The reward is calculated follow:
1. if constraints are violated.
2. $0.5*real (Br*con) (Hr) tconj (Hphi) +kphi+con] +con (Hz) *8z) /Vs, the
+ inductance value the inductor, if constraints are satisfied.
## Example
An example solution is shown below. You should copy the example solution, but you can refer it to understand the task and create better ones.
"yani
geometry:
main:
type: Difference
geometries_at add:
outer:
type: Polygon table:
[0, 0.2]
10.18, 0.2]
10.18, -0.2]
10, -0.21 geometries scometries_subtractt
inner:
type: Folygon
table:
[0.03, -0.155]
[0.03, 0.155]
12, 0.155] 10.12, -0.155]
selection:
core:
type: UnionSelection
geometries:
main
coils:
[0.08, 0.11]
[0.08, 0.09]
[0.08, 0.07]
[0.08, 0.05]
[0.08, 0.03]
[0.08, 0.01]
[0.08, -0.01]
[0.08, -0.03]
10.08, -0.051
10.08, -0.071
10.08, -0.091
[0.08, -0.11]
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Prompt for task Beam Bending
You are a helpful AI Assistant that provices well-reasoned and detailed responses.
You first think about the reasoning process internal monologue and then provide the user with the answer.
Respond in the following format: <think>
</think>
<anawer>
</answer>.
## Task Description
You are helpfu1 Al Assistant and scientist with strong physical background and wonderful geometric designing ideas.
You are asked to generate the geometry design of a component using
yaml files under certain constraints. You will first create geometries of your design, and then assign functions to the geometries according the specific requirements.
Your final answer should contain a yaml file enclosed in ""yaml An (your code) "* The yaml file should have at least two parts: geometry and selection. The specific requirements are as follow:
1. geometry: A list of objects with type and type-specific parameters. The types and parameters are follows:
Polygon: (2D) You
it create rectangles, triangles, etc.
table: Ordered list n vertices as [x, y] points. The + polygon is formed by **connecting consecutive points** (1+1) and **automatically closing** the shape (p_n->p_1). **NO Intersections between edges/nodes are allowed**.
fillet:
(Optional) A list of [i, r] tuples, where i the index (starting from 1) of a polygon vertex defined in the above table, and is the fillet radius for that corresponding vertex.
gllipas: (2D) You can use
create circles.
semiaxes: [horizontal, vertical] axis lengths
pos: [center_x, center_y] center position
rot: (Optional) Rotation angle (degree) counterclockwise
angle: (Optional) Angular span (degree) counterelockwise e.g.
+ by setting angle=180 you can draw upward semicircle.
LineSegment: (1D)
coordl: [start_X, start_y]
coord2: [end_x, end_y]
CircularAre: (1D)
F: Radius
anglel: Start angle (degree) counterclockwise, 0 degree represent positive direction of X-axis. angle2: End angle (degree) counterclockwise
CubicBezier: (1D)
p: Control points as x2, 1y0,y1,y2,y311 w: weight values as [2,,
Interpolatiortuure: (ID)
table: Ordered list of [x,y] points interpolate through.
The curve will pass every points smoothly (polynomial interpolation for x and y).
ParametricCurve: (1D)
parname: Name of paramoter
parmin: Minimum value of parameter
parmax: Maximum value of parameter
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coord: Expressions about the parameter like ["expression "expression_y".. Trigonometric functions here use radians Convertrosolid: (2D) Geometry formed by end-to-end connected 1c curves.
geometries: A dictionary of 1D geometries (using the same structure the top-level gecmetry section, recursive). simply connected
**They Must connect end-to-end and space**.
Union: (2D) Union of 2D geonetries.
geometries: A dictionary of geometries (recursive)
Intersaction: (2D) Intersection of 20 geometries.
geometries: A dictionary of geometries (recursive).
Difference: (2D) Difference of the 2D geometries.
geometries add: A dictionary of geometries to keep (recursive).
add:
geomstries_aubtrac:: dictionary of geometries subtract (recursive).
After **geometry** was created, the shapes will be splitted into ##non-overlapping connected regions**.
Overlapping 2D shapes create new regions (e.g., two intersecting circles regions)
Enclosed 2D shapes split regions (e.g., circle inside polygen
2 regions: circle interior + polygen-ring)
curves through 20 shapes create sub-regions (e.g., line - segment through rectangle alternating regions)
The **regions** can be represented by the following ways:
point: You can select an interior point of the region to
represent it. The point should One point per region suffices.
boundaries/sormers:.
geometry: The 2d shapes you created might be splitted into several regions. You can select the geemetry to represent all the regions in it.
2. selection: After regions are created, you will assign different functions to regions using selections.
UnionSelections Union of all the regions selected below. points: (Optional) List of [x,yl points representing distinct + regions.
geometries: (Optional) List of 2d gecmetry names you created above. By listing geometries here, you can select all the region this geometry contains.
selections: (Optional) List other selection names you created.
+
IntersectionSSelettion Intersection of all the regions selected below.
same parameters UnionSelection
Differenceselection. Select regions Subtract.
not
+
add: same parameters as Uni UnionSelection. subtract: same parameters UnionSelection
Finally a yaml file will like the following sample:
""yani
geometry:
unil: Name of this geometry
type: Union
geometries: # create geometries recursively below uni ell: # Name of the first ellipse to union
type: Ellipse Type of this geometry
semiaxes: 12.0, 1.0) Specific parameters pos: [1.0, 1.0]
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uni poll: Name of the second poly
type: Polygon
table:
[-1.0, -0.3]
[2.0, -1.0]
[1.0, 1.01
linel: This line splits the cllipse into 2 regions. type: LineSegment
coordl: [1.0, 2.0]
coord2: [3.0, 1.0]
selection:
sell: Name of this selection
type: DifferenceSelecttinn
add:
geometries:
unil Select all the regions in unil
subtract:
points:
[2.5, 1.5] # Remove the region where (2.5, 1.5) in. This 3 region is part of ellipse but splitted by the line segment.
## Beam Section Geometry Design
You are asked to design the cross section of beam. The object iveeivve ivve is to maximize both the largest principal moment of inertia and the small.
section The goal can be quantified as (I1++0.8 I2**0.2) /A$ with being the largest principal moment of inertia, I2 being the smallest principal moment of inertia and A being the beam cress section area.
torsional constant, while keeping the
The beam cross-sectin ional dimension should not go beyond 0. 0.15m, with the center staying close to the origin.
You required design the beam have to be symmetric. After you create the gecnetry, you should select the regions of the beam. **The Name of the selection must be "beam**.. **.
section. The shape doesn't
The constraints are as follow:
1. The shape center should be close to the origin.
2. The shape should stay inside the circle boundary with radius 0.2m.
3. The area should not be smaller than 2e-3 m²2.
The reward is caleulated as follow:
1. if constraints are violated,
2. (T1++0.88 12.+0.2) /AS, weighted geometry average of the largest
principal moment of inertia and the smallest principal moment of inertia, normalized by the cross section area, if constraints satisfied.
## Example
An example solution is shown below. You should copy the example solution, but you can refer it to understand the task and create better ones.
yami
geometry:
poll:
type: Polygon
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table:
[0.05, 0.05]
[-0.05, 0.05]
[--000,, 0.04]
[-0.005, 0.04]
[-0.005, -C.04]
[-0.05, -0.04]
[-0.05, -C.05]
[0.05, -0.05]
[0.05, -C.04]
[0.005, -0.04]
[0.005, 0.04]
10.05, 0.04]
fillet:
14, 0.003]
[5, 0.003]
[10, 0.003]
[11, 0.003]
selection:
beam:
type: UnionSelection
geometries:
poll
Prompt for task Magnetic Torque
You are helpful AI Assistant that provides well alll-resooed and detailed responses.
You first think about the reasoning process as an internal monologue + and then provide the user with the answer.
Respond in the following format: <think>
</think>
<answer>
</answer>.
## Task Description
You are a helpful AI Assistant and scientist with strong physical + background and wonderful geometric designing ideas.
You are asked to generate the geometry design of component using
yaml files under certain constraints. You will first create geometries of your design, and then assign functions to the geometries according to the specific requirements.
Your final answer should contain yaml file enclosed in ""yaml An (your code) "* The yaml file should have at least two parts: geometry and selection. The specific requirements are as follow:
geometry: list of objects with type and type-specific parameters. The types and parameters are as follows:
Polygon: (2D) You can use it to create rectangles, triangles, etc. table: Ordered list of n vertices as [x, yl points. The polygon is formed by **connecting consecutive points** (P_ (p_i->p_(i+1) and *wautomatically closing** the shape (p_n->p_ll)
fillet: (0ptional) A list of [i, r] tuples, where i is the
index (starting from 1) of a polygon vertex defined in the
above table, and is the fillet radius for that corresponding vertex.
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Bllipss: (2D) You
it create circles. semiaxes: [horizontal, vertical] axis lengths
pos: [center_X, center_y] center position
rot: (Optional) Rotation angle (degree) counterclockwise
angle: (Optional) Angular span (degree) counterclockwise. e.g. by setting angle=180 you draw upward semicirale. LineSegment: (1D)
coordl: [start_x, start_y]
coord2: [end_x, endyy]
CircularAro: (ID)
r: Radius
anglel: Start angle (degres) cunterclockwise, degree represent positive direction of X-axis.
angle2: End angle (degree) counterclockvise
CubicBezier: (1D)
p: Control points as
x3], [y0,
w: Weight values as [w0, w3]
InterpolationCurve:: (1D)
table: Ordered list of [x,y] points to interpolate through.
The curve will pass every points smoothly (polynomial 4 interpolation for x and y)
Parametriccurve: (1D)
parname: Name of paraneter
parmin: Minimum value of parameter
parmax: Maxi value parameter
coord: expressions about the parameter like ("expression_*' 4 "expression_y") Irigonometric functions ons here use radi ans ConvertIoSolid: (2D) Geonetry formed by end-to-end connected 1D
geometries: A dictionary +
1D gecmetries (using the same structure as the top-level geometry section, recursive). +*They Must connect end-to-end and form simply connected
space**.
Union: (2D) Union
geometrics.
geometries: A dictionary of geometries (recursive).
Intersection: (2D) Intersection of 2D geometries.
geometries: A dictionary of geometries (recursive).
Difference: (2D) Difference of the 2D geometries.
geometries_art add: A diczionary of geometries to keep (recursive).
geometries_oubtract: A dictionary of geometries to subtract (recursive).
After **geometzy** was created, the shapes will splitted into -noo--overlapping connected regions**.
Overlapping 2D shapes create new regions (e.g., two intersecting circles regions)
Enclosed 2D shapes split regions (c.g., circle inside polygen
+ 2 regions: circle interior + polygon-ring)
ID curves through 20 shapes create sub-regions (e.g., line segment through rectangle alternating regions)
The **regions** can be represented by the following ways:
point: You can select an interior point of the region to
+ represent it. The point should never on boundrries/comers: One point per region suffices.
geometry: The 2d shapes you created might be splitted into several regions. You can select geometry to represent the regions in it.
2. selection: After regions are created, you will assign different functions regions using selections. Unionselection: Union of all the regions selected below.
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points: (Optional) List of [x,y] points representing distinct regions.
geometries: (Optional) List of 2d geonetry names you created above. By listing geometries here, you can select all the region this geonetry contains.
you IntersectionSelcctiin Intersection of all the regions selected below.
selections: (Optional) List of other selection created.
same parameters UnionSelection Differanceselecti on: Select the regions Add but + Subtract.
add: same parameters as UnionSelection.
subtract: same paraneters as UnionSelection.
Finally yaml file will like the following sample:
""yani
geometry:
unil: Name of this geometry
type: Union
geometries: # create geometrics recursively below uni_ell: $ Name of the first ellipse to union
type: Ellipse Type of this geometry
semiaxes: [2.0, 1.0] Speci parameters
pos: [1.0, 1.0]
uni poll: $ Name of the second polygon
type: Polygon
table:
[-1.0, -0.3]
[2.0, -1.0]
[1.0, 1.0]
linel: This line splits the cllipse into 2 regions.
type: LineSegment
coordl: [1.0, 2.0] coord2: [3.0, 1.0]
selection:
sell: Name of this selection
type: DifferenceSeleeciionn
add:
geometries:
unil Select all the regions in unil
subtract:
points:
[2.5, 1.5] # Remove the region where (2.5, 1.5) in. This region is part of ellipse but splitted by the line segment.
## Geometric Design of 2D Iron Core
You are asked to design a 2D iron core. The iren core has large permeability and is subject to a constant far field magnetic field
intensity which applies a magnetic torque (pointing out of the 2D The objective is to maximize the magnetic iron core small. The goal can quantified as SIT2| /A$ where T2 is the torque in the out of plane direction and A is the area of the core.
plane) the iron
torque while keeping
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The iron core should be designed inside circle air domain of radius 0.05m centered at the origin 0), The boundary of the circle domain is subject to a constant magnetic field intensity of [0,1e5] A/m.
You are required give the geometry of the iron core. The shape doesn 't have to be symmetric. After you create geometry, you should select the regions of the corc. **The Name of the selection must be core'**.
The constraints
follow:
1.
The core center should be close to the origin and inside the circle air domain of radius 0.02m centered at (0,
+
2. The boundary of the core should stay at least 0.02m away from the circle air domain.
3. The Area of the core should between 2e-4 and 2e-3 m.
The reward is calculated follow:
1. 0 if constraints are violated.
2. $ITZI/AS, the absolute value of magnetic torque generated by the
+ constant far field magnetic field intensity on the iron core normalized by the iron core area, if constraints are satisfied.
Example
example solution is shown below. should copy the example aolution, but you can refer to it to understand the task and create better ones.
yani
geometry:
poll:
type: Polygon.
table:
[0.01, -0.003]
[0.02, 0.001]
[0.01, 0.01]
[-0.01, 0.005]
1-0.02, -0.008]
120.02, -0.008]
1-0.003, -0.01|
selection:
core:
type: Unionselection
geometries:
poll
Prompt for task Periodic Heat
You are helpful AI Assistant that provides detailed responses.
and
You first think about the reasoning process as an internal monolcgue + and then provide the user with the answer.
Respond in the following format: <think>
</think>
<anawer>
</answer>.
# Task Description
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You are helpful AI Assistant and scientist with strong physical background and wonderful geometric designing ideas.
You are asked to generate the geometry design of component using
yaml files under certain constraints. You will first create geometries of your design, and then assign functions to the geometries according the specific requirements.
Your final answer should contain a yaml file enclosed in ""yaml An (your code) The yaml file should have part named follow:
geometry. The specific requirements
1. geometry: 1ist of objects with type and type-specific fis parameters.
The types and parameters are as follows:
Polygon: (2D) You can use it to create rectangles, triangles, etc. table: Ordered list
vertices [x, yl points. The polygon is formed by **connecting consecutive points**
+ (p_i->p_(1+1) and **automatically closing** the shape (p_n->p_1).
fillet: (Optional) A list of [i, r] tuples, where i is the defined in the above table, and r is the fillet radius for that corresponding vertex.
index (starting from of a polygen
Ellipse: (2D) You can use it to create circles. semiaxes: [horizontal, vertical] axis lengths
pos: [center_x, center_y] center position
rot: (Optional) Rotation angle (degree) counters
angle: (Opti Angular span (degree) countercl conterrlockwissee se. e.g. by setting angle=180 you can draw upward semicircle.
LineSegment: (1D)
coordl: [start_x, x, start_y]
coord2: [end_x, end_y]
CircularAre: (1D)
Radius
anglel: Start angle (dagree) counterclockvise, degree
+ represent positive direction of X-axis.
angle2: End angle (degree) counterclockvise
CubicRezier: (ID)
p: Control points
x1,0, yy0,,1,y2,y311
w: Weight values as ww0, w3] InterpolationCurver (1D)
table: Ordered list of [x,y] points to interpolate through.
The curve will pass every points smoothly (polynomial
interpolation for and y).
PerametricCurve: (1D)
parname: Name of parameter
parmin: Minimum value of parameter
parmax: Maximum value of parameter
coord: Expressions about the parameter like ["oxpression_x* + "expression_y"). Trigonometric functions here use radians ConvertToSolid: (2D) Geometry formed by end-to-end connected 1D curves.
geometries: dictionary 1D geometries (using the structure as the :op-level gecmetry section, recursive). *+They Must connect end-to-end and form simply connected
space**.
Union: (2D) Union of 2D geometries.
geonetries: A dictionary geometries (recursive).
Intersection: (2D) Intersection of 2D geonetries.
geometries: A dictionary of geometries (recursive).
Difference: (2D) Difference of the 2D geometries.
geometries_ade A dictionary geometries keep (recursive).
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geometries_eubrrace dictionary of geometries subtract (recursive).
## Structure Design of the 3D heat transfer unit cell
You asked design unit cell structure in 3D. The objective to maximize the effective thermal conductivity with limited
+ matorial usage, which can be quantificd as Strace (k_[cff)) A where sk_(eff1 is the effective
thermal conductivis iv matrix of shape 3+3, and rho_eff is the effective dersity.
You should first define a 2D rectangular unit cell domain by giving the width and height of the domain.
You then design the hollow part of the 2D unit cell. You must create a geometry named 'hollow', represents the hollow part of the unit cell. Four copies of this geometry object will be created by translating the original object with the following vectors [=cell_width/2, -cell_height/2), F-cell_width// cell cell_height/2], [cell_width/2, -cell_height/2)) [cell_width/2, cell_ sill_height/2].
The unit cell domain will be subtracted by the geonetry object and its
copies. The subtracted areas are filled with air (thermal conductivity "0.026W/ (m K), density "1.174kg/m"3) and the remaining areas are filled with aluminum (thermal conductivity 238w/ (m K), density 2700 kg/m²3).
The final 3D unit cell structure is generated by extruding the 2D unit cell.
The unit coll will subject to periodic boundary condition in x, y, and
2 directions. Your design should provide higher effective thermal conductivity using reasonable amount of aluminum and carefully designed hollow shapc
The constraints are as follow:
1. The original geometry object ("hollow") should not overlap with the boundary of the domain. But its copies may overlap with the boundary.
2. The original and copied geonetry objects should not overlap or adjacent with each other.
3. The effective density should not exceed 2000 kg/n'3. You should control the usage aluminum (by create larger hollow parts).
4. The unit cell domain is centered strictly at the origin. The geometry you designed should be centered approximately at the origin, as it may not be symestrcc.
The reward is caleulated as follow:
1. if constraints are violated,
2. Strace (k (k_eff) /rho_efff, the effective thermal conduativity of the unit cell structure normalized by the effective density, if constraints
satiafied.
## Example
An example solution is shown below. You should not copy the example solution, but you can refer to it to understand the task and create better ones.
"yami
cell:
sizes: [5e-3, 3e-3]
geometry:
hollow:
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type: Polygon
table:
[1. 61e-3, 0]
-8804e-4, 1.45e-3]
[-8.00e-4, 1.45e-3]
-[-1.6le-3, [-8.000-4, -1.45e-3]
18.04e-4, -1.450-3]
Prompt for task Demultiplexer
You are a helpful AI Assistant that provides well -ressorddddd and detailed responses.
You first think about the reasoning process internal monologue and then provide the user with the answer.
Respond in the following format: <think>
</think>
<answer>
</answer>.
## Task Description
You are a helpful AI Assistant and scientist with strong physical + background and wonderful geometric designing ideas.
You are asked to generate the geometry design of component using
yaml files under certain constraints. You will first create geometries of your design, and then assign functions to the geometries according to the specific requirements.
Your final answer should contain yaml file enclosed in ""yaml (your code) The yaml file should have at least two parts: geometry and selection. The specific requirements are as follow:
1. geometry: A list of objects with type and type-specific parameters. The types and parameters
follows:
Polygon: (2D) You can use it create rectangles, triangles, etc.
table: Ordered list of n vertices as [x, yl points. The polygon is formed by **connecting consecutive points** (p_i->p_(++}}) and vwautomatically closing** the shape (p_r->p_1).
fillet: (Optional) A list of li, rl tuples, where i the
index (starting from 1) of a polygon vertex defined in the
above table, and is the fillet radius for that corresponding vertex.
Bllipse: (2D) You can use
create circles. semiaxes: [horizontal, vertical] axis lengths
pos: [center_x, x, center_y] center position
rot: (Optional) Rotation angle (degree) counterclockwise
angle: (Optional) Angular span (degree) counterclockwise e.g. 4 by setting angle=180 you can draw a upward semicircle. LineSegment: (1D)
coordl: [start_x, start_y]
coord2: [end x, end
CircularAra: (1D)
r: Radius
anglel: Start angle (degree) counterclockwise, 0 degree represent positive direction of X-axis. angle2: End angle (degree) counterclockwise
CubicBezier: (1D)
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p: Control points x2, x31, y31]
w: Weight values [20 w3]
Interpolationturve. (1D)
table: Ordered 1ist of [x,y] points to interpolate through.
The curve will pass every points smoothly (polynomial
interpolation for and y).
Parametriccurve: (1D)
parname: Name of paraneter
parmin: Minimum value of parameter
parmax: Maximum value of parameter
coord: Expressions about the parameter like ["expression_ rexpression_y"]. Trigonometric funct ions here use radians ConvertIoSolid: (2D) Geonetry formed by end-to-end connected 1D curves.
geometries: A dictionary ID gecmetries (using the structure as the top-level geometry section, recursive).
4 **They Must connect end-to-end and form simply connected space**.
Union: (2D) Union of 2D geonetries.
geometries: A dictionary geometries (recursive).
Intersection: (2D) Intersection
2D geometries.
geometries: A dictionary of geometrics (recursive).
Difference: (2D) Difference of the 2D geometries.
geometries add: A dictionary geometries to keep (recursive).
geometries_subtran (recursive).
dictionary of geometries to subtract
After **geometxy** created, the shapes will splitted into +nnon-overaapping connected regions**.
Overlapping 2D shapes create new regions (e.g., two intersecting circles - 3 regions)
Enclosed 2D shapes split regions (e.g., circle inside polygen
regions: cirele interior + polygon-ring)
curves through 2D shapes create sub-regions (e.g., line segment through rectangle + alternating regions)
The **regions** can be represented by the follewing ways:
point: You
select interior point of the region
+ represent it. The point should never on boundaries/corners. One point per region suffices.
geometry: The 2d shapes you created might be splitted into several regions. You select the geometry to represent all the regions in it.
2. selection: After regions are created, you will assign different functions to regions using selections.
UnionSelection: Union of all the regions selected below.
points: (Optional) List + regions.
[x,y] points representing distinct
geometries: (Optional) List of 2d geometry names you created above. By listing geometries here, you select all the region this geometry contains.
selections: (Optional) List other selection names you created.
IntersectioonSeleetion Intersection of all the regions selected below.
same parameters as UnionSelection
Differenceselectinn. Select the regions in Add but not in Subtract.
add: same parameters UnionSelection. subtract: same parameters UnionSelection.
Finally a yaml file will be like the following sample:
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yaan geometry:
unil: Name of this geometry
type: Uniin
geometries:
create geometries recursively below el1: & Name of the first ellipse union
type: Ellipse Type of this geometzy
semiaxes: 12.0, 1.0) Specific parameters
pos: [1.0, 1.0]
uni poll: Name of the second polygon
type: Polygon
table:
[-1.0, -0.31
12.0, -1.01
[1.0, 1.0]
linel: This line splits the ellipse into 2 regions.
type: LineSegment
coordl: [1.0, 2.0]
coord2: [3.0, 1.0]
selection:
sell: & Name of this selection
type: DifferenceSelectiinn
add:
geometries:
unil # Select all the regions in unil
subtract:
points:
12.5, 1.5] $ Remove the region where (2.5, 1.5) in. This - region is part of cllipse but splitted by the line segment.
## Geometric Design of a sound wave demultiplexer
You are asked to design 2D sound wave demultiplexer. The from port and omits sound wave at port 2 and The objective is to maximize the difference of sound pressure (on log scale) at two outlet ports, (port3.P_(outt with $P_(out) being the sound pressure the outlet ports.
demultiplexer takes incident sound
which is calculated as $log10 (ooort.._(out))-DDi
The entire pressure acoustic region will be a circle of radius 0.1m centered at (C,0) The incident wave comes from the negative x-direction (9 o' olock).). The sound waves are then ommited at 1 o'clock (port 2) and 5 o'clock (port 3) of the acoustic region.
You should design void geometry (material to be removed) so that the sound wave will propagate through the remaining geometry and maximize the objective function. You should create 1ist of basic geometries and then select from them form the void regions. **The Name the selection must 'void'**. Keep mind that the void geometry should stay inside the acoustic region and at least 0.15m away from the boundary of the acoustic region.
The constraints are as follow:
1. After removing the void geonetry, the remaining part should still be connected.
2. The void geometry should stay inside the accustic region, and at least 0. 02m away from the boundary of the acoustic region.
The reward is calculated as follow:
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1. if constraints are violated. $1og10
the log scale pressure - difference between port 2 and port 3, if constraints are satisfied.
## Example
An example solution is shown below. You should not copy the example solution, but you can refer to it to understand the task and create better ones. Feel free to add more basic geometrics.
"yaa""
geometry:
barrier:
type: ConvertToSolid
geometries:
cir_inner:
type: CircularAre
r: 0.08
anglel: -120
angle2: C
cir cir_outer:
type: CircularAre
r: 0.06
anglel: -120
angle2: C
linel:
type: li lineSegment
coordl: [0.06, 0.01
coord2: [C.08, 0.01
line2:
type: LineSegment
coord1: 10.06xcos (-120+pi/180), 0.06+sin(-120pp//1181
coord2: [C.08 10.08+co0 (-120+p1/180),
G.08-sin(-120-p1/88)
selection:
void:
type: UnionSelection
geometries: [barrier]
D.3 CIRCLE PACKING
Prompt for task Circle Packing
You are an expert software developer tasked with iteratively improving + a codebase.
Your job is to analyze the current program and auggest improvements
based feedback from previous attempts. Focus on making targeted changes that will increase the program's s + performance metrics.
Respond in the following format: <think>
</think>
<anawer>
</answer>.
Problem Description
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You are expert mathematician specializing in circle packing problems and computational geonetry. Your task is to improve constructor function that directly produces a specific arrangement of 26 circles in a unit square, maximizing the sum of their radii. The AlphaEvolve paper achieved a sum of 2.635 for n=26.
Key geometric insights:
In dense regions, circles often follow hexagonal packing patterns, with the theoretical maximum density for infinite packing being pi/ (2sqrt -3.))0.906)
However, when confined perfect symmetry and make pure hexagonal packing suboptimal.
finite square, **edge effects** éisrupt
Optimal arrangements often require svariaabe-sized circles**, as this can improve space utilization compared to equal radii. Larger circles can be placed toward the center, with smaller circles strategically fitted near edges
corners.
Effective designs may use **layered or shell-like patterns** rather than strict hexagonal grids. Hybrid approaches combining regular arrangements in dense regions with adaptive adjustments near boundaries lrre common in the densest known packings.
The #roptimization method** plays a critical role: physics-inspired simulations or algorithms with well-tuned parameters can yield better configurations than purely geometric intuition.
Mathematical research indicates that for certain specific values achieve unusually high densities.
n, special arrangements
You may either designing an explicit constructor of the result or explore sasrdh-based, optimization, even multi-stage optimization methods, long they finish running within minutes.
## Current Program
Status: [current_status]
"python (current_program)
## Task
Suggest improvements the program that will Lead to better performance on the specitied metrics.
You MUST the exact SEARCH/REP diff format shown below to indicate changes:
<<<α<<< SEARCE
Original code to find and replace (must match exactly)
New replacement code
>>>>>>> REPLACE
You can suggest mu multiple changes. Rach SEARCH section must **xactly** match code in the current program.
Be thoughtful about your changes and explain your reasoning thoroughly.
Make your rewritten program still contains "construct_packing
function and maintains the same outputs. **You add/delete/modify other functions arbitrarily. **
If you want to use new packages, please import them. Usable packages: scipy, sympy, shapely, pulp, evxpy, nlopt, deap
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If your code's execution time exceeds 1 minute, you will receive 0 reward. Pay attention to the runtime efficiency!
IMPORTANT: Do not rewrite the entire program focus on targeted improvements.
D.4 FUNCTION MINIMIZATION
Prompt for task Minimize Function
Problem Description
You expert in optimization algorithms. Your task is to improve function minimization algorithm that minimizes complex non-convex function with multiple local minima. The function is defined in (dimension) -idaiiiiiinn space with the following expression:
"python
(formula]
## Current Program
status: [current_status)
"'python (current_program)
## Task
Suggest improvements to the program that will lead to better performance the specified metrics.
Your code's execution time should not exceed seconds. Pay attention to the runtime efficiency!
You MUST the exact RARCCCE diff format shown below to indicate changes:
<<<<<<< SEARCH
Original code to find and replace (must match exactly)
New replacement code
>>>>>>> REPLACE
Performance is evaluated using:
value_score: Closeness to minimum function value: Iglobal_min|
+ (Iglobal_mini | found_valuee global_minl)
2. distance_scom score: Proximity to true solution point: 1/ (1 +
Blsttaccett__gloal__mg
3. standard_deviation score: Solution stability across runs:
0 (1/(1tsdd_x1) 1/ (1+std [x)_dxx)) ...)/dimm
4. spord_score: Execution efficiency: min imwpaaggnntttteii__eeeeeeeeeeeee 101/10
5. reliability
succesaful_runsttotttoo
Successful has
no tracebacks and timeouts.
6. combined_score **This is the final reward you received. 1008 value_score.
If you
packages, please import them.
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Make your rewritten program still contains "run_search () function and maintains the same outputs. You can add/delete/mociify + other functions arbitrarily.
IMPORTANT:
Do not rewrite the entire program focus on targeted improvements.
D.5 SYMBOLIC REGRESSION
Prompt for Chemistry tasks
You are an expert software developer. Your job is to write Pythen + function based on feedback from previous attempts.
Write your code in exactly the following format:
"python your code
Your code's execution time is limited, so pay attention to runtime efficiency!
If you new packages, please import them.
Ensure the program still contains the func function and produces the same outputs; other functions can be added, deleted, or modified freely.
IMPORIANT: The current task is symbolic regression problem. Write Python expression
func where parameter scales are similar as possible (use linear scaling or translation if needed). This helps later optimization when all parameters are initialized randomly in [C,1].
Respond in the following format: <think>
</think>
<answer>
</answer>,
Your task to evolve a Python function "func (x, params)" that models + a scientific process, considering the physical meaning and relationships of inputs, by predicting output variables based input variables.
The function signature is:
''python
def func (x: np. ndarray,,, params: ndarrayy np.ndarray:
"x" is 2D NumPy array of shape " (n sample,, 2)
"params" is 1D NumPy array of up to 10 parameters
The function should return 1D NumPy array of predictions with + shape (n_samples,)
**Curennt Problem:**
Model the dt_dt (Rate of change of concentration in chemistry reaction + kinetics) using the input features: t (Time), A (Concentration at time t)
Thus, "x" contains 2 columns: t (Time), (Concentration at time t).
The initial version of "func" is simple linear model. Parameters in 'params" will be optimi zed externally using the BFGS algorithm based on unseen training data.
Your objective is evolve "func" improve predictive performance on unseen data. Aim for a balance between:
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*cccuaacy*:: Lower mean squared error (MSE) training data **Simplicit+*:: Frefer concise, interpretable expressions
Model performance (score = -log_
10 (mse)) will be evaluated on a held-out dataset. Ensure the model is free of potential numerical (e.g., log0, division by 0).
## Current Program
Status: Initial Program
""python
def func (x, params)
**" Calculates the model output using a linear combination of input variables
or constant vaiue if no input variabies. Operates matrix of samples.
Args:
x (np. :np.raarra): A 20 numpy array of input variable values, shape (n_sampies, n f_features).
features is 2.
If n features is 0, x should be shape 4 (a_aamples, 0).
The order of columns in x must correspona to: (t, A)
numpy array of parameters. params (np.ndarray): expected length; 10.
Returns:
up.ndarray: A numpy array predicted output values, shape (n (n_sampies,)
result xl;, 0] params [0] + x(i, 1] * params [1]
return result
Prompt for Biology tasks
You are an expert software developer. Your job is to write a Python function based on feedback from previous attempts.
Write your code in exactly the following format:
** "python
your code
Your code's execution time is limited, pay attention to runtime efficiency!
If you use new packages, please import them.
Ensure the program still contains the func() function and produces the same outputs; other functions can be added, deleted, or modified freely.
IMPORTANT: The current task is symbolic regression problem. Write Python expression as possible (use linear scaling or translation if needed). This helps later optimization when all parameters are initialized randomly in [C,1].
func() where parameter scales are as similar
Respond in the following format: <think>
</think>
<answer>
</answer>.
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Your task is evolve Python function "func (x, params) that models scientific process, considering the physical meaning and + relationships of inputs, by predicting output variables based on input variables.
The function signature is:
"'pyhhon
def func np. ndarray, parans: np. ap.nddrrry) -> np .darra:
'x" is 2D NumPy array of shape (n_samples, 2) "pazams" is a ID NumFy array of up to 10 parameters
The function should return 1D NumPy array of predictions with shape (n (n_samples,)
Curreett Problem:**
Model the dP_dt (Fopulation growth rate) using the input features: (Time), P (Population at time t) Thus, "x" contains columns: (Time), (Population time t).
The initial version of "func" is a simple linear model. Parameters in "params" will be optimized externally using the BFGS algorithm based unseen training data.
your objective is to evolve "func" to improve predictivs ive performance on unseen data. Aim for a balance between: **Accuracy**: Lower mean squared error (MSE) on training data **Simplicity*:: Prefer concise, interpretable expressions
Model performance (score = -log_10 (mse)) will be evaluated on a held-out dataset. Ensure the model is free of potential numerical errors (e.g., log0, division by 0).
## Current Program
Status: Initial Program
pyyhon
def func params)
#** Calculates the model output using linear combination of input variables
or constant value if no input variables. Operates on a matrix of samples.
Args:
x (np. :ddrray)) A 2D numpy array of input variable values, shape (n_sampies, n feaauuure)).
A features is 2.
0, x should shape If foaturs + (a_samples, 0).
The order of columns in x must correspond to: (t, P).
params (np.ndarray): Expected length: 10.
numpy array of parameters.
Returns:
np.ndarray: A 10 numpy array of predicted output values, shape (n_samples,).
result x|:, 0] params [0] x[:, 1] params [1]
return result
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Prompt for Physics tasks
You are an expert software developer. Your job to write a Pythen function based on feedback from previous attempts.
Write your code in exactly the following format:
"python your code
Your code's execution time is limited, so pay attention to runtime efficiency!
If you use packages, please import them.
Ensure the program still contains the func function and produces the same outputs; other functions can be acded, deleted, or moaified freely.
IMPORTANT: The current task is symbolic regression problem. Write Python expression
similar as possible (use linear scaling or translation if needed). This helps later optimization when all parameters are initialized randomly in [0,1].
func() where parameter scales
Respond in the following format: <think>
</think>
<answer>
</anawer>.
Your task to evolve a Python function "func (x, params)" that models + a scientific process, considering the physical meaning and relationships of inputs, by predicting output variables based
input variables.
The function signature is:
"python
def func (x: adarray,,,, params: adarray) ndarrayyy::
"x" is 2D NumPy array of shape " "(n_samples, 3)
"params" is 1D NumPy array of up to 10 parameters
The function should return ID NumPy array of predictions with
+ shape (n_samples,)
**Current Problem:**
Model the dv_ (Acceleration in Nonl-linear Harmonic Oscillator)
using the input features: x (Position at time th, t (Time),
(Velocity at time t)
Thus, "x" contains 3 columns: x (Position at time t), (Time), v (Velocity time t).
The initial version of "fune" is a simple linear model, Parameters in "params" will be optimized externally using the BEGS algorithm based unseen training data.
Your objective is to evolve "func" improve predictive performance on unseen data. Aim for a balance between: **Accuracy**: lower mean squared error (MSK) on training data **Simplicitye:: Prefer concise, interpretable expressions
Model performance (score -log_ 10 (mse)) will be evaluated on a held-out dataset. Ensure the model is free of potential numerical errors (e.g., log0, division by 0).
Current Program
Status: Initial Frogram
""python
def func (x, params)
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#*"
calculates the model output using linear combination of input + variables
or constant vaiue if no input variables. Operates on a matrix of samples.
Args:
x (np, :ddaray)) A 2D numpy array of input variable values, shape (n_sampies, r_fcatures).
features is 3.
0, should be shape If n_ features + (n_samples, 0)
The order of columns in x must correspona to: (x, t, v).
params (np.ndarray): 1D numpy array of parameters.
Expected length: 10.
Returns:
np.ndarray: A ID numpy array of predicted output values, shape (n_ v__samiess)
#*" result x(;, 0] params [0] x(i, 1] * parans [1] + x(i, 2]
params [2] return result
Prompt for Material Science tasks
You expert software developer. Your job is to write Pythen function based on teedback from previous attempts.
Write your code in exactly the following format: ""python your code
Your code's 's execution time is limited, so pay attention to runtime + efficiency!
If you use new packages, please import them.
Ensure the program still contains the func function and produces the same outputs; other functions can be added, deleted, or modified freely.
IMPORTANT: The current task is a symbolic regression problem. Write Python expression in func () where parameter acales are simila lar as possible (use linear scaling
translation if needed). This helps later optimization when all parameters are initialized randomly in [0,1].
Respond in the following format: <think
</think>
<answer>
</answer>.
Your task evolve Python function "func (x, params) that models a scientific process, considering the physical meaning relationships of inputs, by predicting output variables based on input variables.
The function signature is:
ppython
def func (x: rdarray, parans: np. ndarray) -> np ndarray:
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'x" is 2D NumPy array of shape (nn_samples, 2)
"params" is NumPy array up to 10 parameters
The function should return a 1D NumPy array of predictions with shape (n_samples,)
Current Problem:**
Model the sigma (Stress) using the input features: epsilon (strain), (Temperature)
Thus, "x" contains 2 columns: epsilon (Strain), T (Temperature).
The initial version of 'func" is simple linear model. Parameters in + "params" will be optimi zed externally using the BFGS algorithm based on unseen training data.
Your objective is evolve "fune" improve predictive performance
on unseen data. Aim for a balance between: **Accuracy**: Lower mean squared error (MSE) on training data
Siimplicitye::: Frefer concise, interpretable expressions
Model performance (score -1og_10 (mse)) will be evaluated + held-out dataset. Ensure the model is free of potential numerical errors (e.g., log0, division by 0).
## Current Program
status: Initial Frogram
ppyhho
def func (x, params):
Calculates the model output using linear combination of input variables
or constant value if no input variables. Operates en a matrix samples.
Args:
x (np. :op.ndarray): A numpy array input variable values, 4 shape (n_ (n_samples, n f_
n_ features is 2.
If n features is 0, x should be shape sampaees 0)
The order of columns in x must correspona to: (epsilon, I).
params (np.ndarray): A 10 numpy array of parameters. Expected length: 10.
Returns:
ap.ndarray: A 1D numpy array of predicted output values, shape (n mn_smmpies,).
result xli, 0] params(O) x(i, params [1)
return result
E METHODOLOGICAL CHALLENGES AND COMPARATIVE ANALYSIS OF RL-EA INTEGRATION
This appendix details the specific technical challenges associated with integrating Reinforcement Learning (RL) and Evolutionary Algorithms (EAs). We further analyze why a naive sequential com- bination (c.g., AlphaEvolve) fails to scale effectively compared to the proposed HELIX framework, supported by empirical evidence from the Circle Packing problem.
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E.1 TECHNICAL CHALLENGES AND SOLUTIONS
The integration of RL and EAs presents several non-trivial challenges, primarily stemming from the conflicting objectives and operational domains of the two paradigms. HELIX addresses these as follows:
Goal Mismatch and Unification. fundamental disconnect exists between RL, which learns policy mapping states to actions, and EAs, which act as population- -based optimization methods relying on recombination and mutation. Integrating these requires principled bridge rather than naive combination.
In-Context Learning as a Bridge: HELIX adopts an in-context learning paradigm where previously discovered high-quality solutions are injected into the prompt explicit mem- ory. This transforms the Large Language Model (LLM) into a parameterized mutation operator, conditioned historical trajectories.
Unified Optimization: We employ Group Relative Policy Optimization (GRPO) to train this mutation operator. GRPO naturally generates diverse rollouts that serve as the popu- lation for evolutionary selection. Consequently, policy optimization (RL) and evolutionary search (EA) are coupled within a closed loop: test-time scaling via evolution provides high- quality data for RL, while RL improves the mutation operator for subsequent evolutionary steps.
Diversity Estimation in Giant Code Spaces. Traditional evolutionary metrics are ill-suited for code optimization, where the action space is discrete, high-dimensional, and highly structured. Measuring individual diversity this domain is challenging. HELIX resolves this by utilizing an embedding- based approach to quantify semantic distances between code individuals. We compute population diversity via k-nearest neighbors (kNN) this embedding space, providing scalable and semantically meaningful metric to guide selection.
E.2 LIMITATIONS OF NAIVE INTEGRATION: A CASE STUDY
To demonstrate why HELIX offers a necessary advancement over "naive" integration, we compare it against the AlphaEvolve paradigm. AlphaEvolve represents sequential approach: post-training an LLM on general domains followed by applying evolutionary algorithms to downstream tasks without further policy updates.
We conducted a comparative experiment on the Circle Packing problem (maximizing the sum of radii for 26 non-overlapping circles in unit square). We evaluated Direct Prompting (BO64), OpenEvolve (an open-source reproduction of AlphaEvolve), and HELIX using Qwen 14B and 32B base models.
Table 2: Performance Comparison on Circle Packing Task
Method
Score (Sum of Radii)
Direct Prompt (Qwen 14B)
1.673
OvenEvolve (Qwen 14B)
1.586
OvenEvolve (Qwen 32B)
1.956
HELIX (Qwen 14B)
2.636
As shown in Table [] OvenEvolve with Qwen 14B performed worse than the Direct Prompt baseline, despite utilizing significantly more computational resources. Our analysis identifies two critical failure modes in naive integration of OpenEvolve:
Naive approaches are heavily constrained by their initial seed solutions. OvenEvolve generates a small set of seed trials (e.g.. 5) and iterates upon them. If these initial trials lack diversity occupy low-performance region, the evolutionary process stagnates in local optima. In contrast, Direct Prompting (BO64) benefits from 64 i.i.d. evaluations, offering broader initial coverage that the naive evolutionary process failed to surpass.
1. Constraints of Initialization Bias.
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2. Rejection of Novelty and Destructive Changes. A more subtle failure mode is the jection of potentially high-reward strategies that require initial "destructive" changes. In our Qwen 32B OpenEvolve experiment (4,147 trials), the model predominantly attempted to ad- just coordinates directly. Only 12 trials (0.3%) attempted a radically different approach using sci py opt i mi ze. mi minimize.
The Failure:
All 12 trials initially yielded a reward 0.0 due to minor compilation errors timeouts. Traditional evolutionary selection, driven by immediate reward superficial code features (e.g., length), discarded these candidates.
The Consequence: The system failed explore the scipy approach, which once de- bugged-ip is capable of yielding scores > 2.0.
E.3 THE HELIX ADVANTAGE
HELIX overcomes the aforementioned limitations through two specific mechanisms:
1. Explicit Diversity Accounting: By using an embedding model to distinguish methods semantically, HELIX assigns a high Diversity Score to the rare scipy-base solutions, even when their immediate reward is low. This ensures they retained in the population for further mutation/deeugginng..
2. Parameter Learning via RL: Once diversity-preserved rollout successfully fixes the implementation bug (generating high-reward solution $1+1), HELIX utilizes this trajec- tory for RL updates. This update increases the probability of the policy generating similar sophisticated methods in future steps.
This establishes a positive feedhack loop: diversity metries preserve potential innovation, and RI. consolidates successful realizations of that innovation into the model parameters, allowing HELIX to break out of local optima where naive methods stagnate.
THEORETICAL ANALYSIS OF THE FRAMEWORK
In this section, employ simplified mathematical model to provide theoretical insights into the advantages of our algorithm in solving complex scientific problems. We demonstrate the efficiency of HELIX in discovering optimal solutions compared baseline methods.
F.1 MATHEMATICAL SETUP AND PRELIMINARIES
First, we establish the geometric and probabilistic foundations of the problem. For a given problem q. we assume the existence of the following structures:
Solution Space: A set of solutions S, which can be viewed as simply connected open manifold complex space.
Reward Function:
A continuous function R: R
Embedding: A mapping & S + TR. that maps the solution space to an Embedding Space
RR" satisfying:
1. Continuity: For any 81, 82 E S derived via similar methods, their embeddings ((s)) and U/2 = (82) are adjacent in R.
Injectivity: Distinct solutions have distinct embeddings. Open Map: maps open sets in S to open sets in R
We define the reward function in the embedding space R follows. For any E R":
R (v)) 0
E 4(S) e ((S)*
r(v)
(29)
where Φ -- (v) is the solution corresponding to Restricted to the image set, T S ((S) is a bijection, making its inverse well-defined. Given the continuity & and R, and the open mapping property of 4, r(v) is continuous.
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Definition F.1 (LLM Transition Process). For solution an LLM parameterized by 0 transforms the solution by outputting an action a ~ (e(-(s) [s). resulting in s' = T(s, a). Based on this, we define the LLM Transition Function I a stochastic process:
TØ RR (92 ++R" >"), To ( (o), where x0(-100 (v)).
(30)
For tractability, we approximate To as an independent Normal distribution:
To ~ Ni )o(v), oT).
(31)
This implies each transition follows v - + 6)(w) + &. where & ~ ol), This Gaussian approximation is justified LLMs typically generate modest modifications the current solution, making local approximations valid in the embedding space.
F.2 THEORETICAL ANALYSIS GRPO
F.2.1 SETUP AND ASSUMPTIONS
In the GRPO method, since the prompt is fixed, the model evolves solely from an initial solution vg. The transition samples from N(es + 6p(vo), GRPO estimates the gradient of the reward function near v = Up + (1))) and updates the model parameters. The effective update dynamics in the embedding space be described as:
(v() 60(v0) Ver + dg(vo)),
(32)
which simplifies to the gradient ascent process e + v vr(),, where yy is the learning rate.
To analyze convergence, we introduce the following assumption regarding the reward landscape.
Assumption F22 (Reward Landscape). We assume the reward function r(u) consists of two Gaus- sian peaks, representing a local optimum (Vioe) and a global optimum (Vopt):
r(t)= Aioc exp (- [Ill ||u-[toell ²
||e (-lle- Vopt 11
Aopt exp
(33)
2w
2w
Let Vopt - Uloell be the distance between the optima.
Theorem F.3 (Convergence to Local Optimum of GRPO). Let vn be the initial solution for GRPO. GRPO will converge to the local optimum near Viec if the following conditions are met:
1. Separation: L 2w. There is sufficient separation between the global and local optima.
2. Amplitude: Asoc > Aopt 1 exxp (-
L²²)
The local optimum is not significantly weaker 2u
than the global optimum locally.
3. Initialization: Decomposing the initial solution as Vo Uloc U + + - Uloe). where
V1 L (Vopt Vioc). we require "to "booriirr 1/2 7,5 Aioc This implies the initial solution is geometrically closer to the local optimum's 's basin of altraction.
w² In Aopt,
Comment. These assumptions hold in many scientific problems where distinct methods (local vs. global) have a large semantic gap (L > 2w), and initial "naive" solutions naturally fall closer to simpler local optima. This illustrates that GRPO, lacking memory or population mechanisms, is prone to trapping local optima.
F.2.2 PROOF OF THEOREM
We decompose the gradient of r(v). Let be parameterized Utoe + (Uopp Uloe). The gradient Vr(v) satisfies:
Atoe (Vioe v) exp (- ||I Utoell
Aapt (- || Uopel 113
Vr(u)
(Nopt u) exp
(34)
2
2
2
2w
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Projecting onto the line connecting the optima (Vopt Vloe):
Alo
(- (_221. 1.2 1111
(Vr(u), Vopt (loc)
yexp
w²
2w
Aope
(1 - 2)2 L2 + 1111
(1 (1-7)en exp
(35)
w²
2w
Projecting onto the perpendicular component UL:
(Vr(o),e1) |/u
112
||v Ulocl 12
(-lu- -oppt 11 ))
(Arcc
+ Aopt
(36)
w²
2w
2w12
First, analyzing the dynamics of v1:
d
0 2 2(01, (1) -((u)|eL
(37)
dt
Regardless of initialization, Ti decays exponentially to The system converges to the linear mani- fold connecting Uloc. and Vopt- Assuming V1 0, the dynamics of y are governed by:
dy x Atoc exp (- 22,
(-(1-11)) (1-17))21²
+ (1 - 7)) exp
(38)
2w
2w
Solving for equilibrium points
( 0) yields stable local equilibrium near 7 an unstable
saddle point Mborrier R 2 w the system flows to the local optimum.
In Aopt and stable global equilibrium near y 1. If < Yborriers
L Aioe"
F.3 THEORETICAL ANALYSIS EVOLVE AND HELIX
F.3.1 SETUP: UNIFIED DRIFT- DIFFI--IFFSION AND SELECTION FRAMEWORK
We analyze the iterative processes of Evolve and HELIX by modeling them as continuous-time stochastic processes. Both algorithms maintain population P and update it via v'.
Evolve (Selection-Diffusion))
In the Evolve algorithm, the model parameters cannot be adjusted. Thus, we assume the model has no inherent directional bias towards different methods of this specific problem (ōp(v) 0). The iteration simplifies + of is generated.
y' = e + aE. At each step, solution is drawn from P. and Critically, solutions with higher Reward are sampled with higher probability. We can model this selection by a weight function w(v) = exp(crr (c)())) where a represents the selection pressure. The new solution is added the population: < (u').
HELIX Difift-Diffusion): HELIX maintains a population and dynamically adjusts directional bias óg (v). Through the GRPO mechanism, this direction will gradually approx- imate the gradient Vr(v). Upon sufficient convergence, the HELIX iteration approximates guided random walk: t' + v)(rv) + of. In HELIX. also sample high-Roo Reward solutions with higher probability, but for mathematical tractability, we assume the selection weight parameter 0. meaning the sampling weight is uniform (w(v) 1).
The comparison is summarized in Table
Table 3: Comparison of Algorithm Dynamics
Algorithm
Dynamics Equation
Drift D(u)
Selection w(v)
Evolve HELIX
J' = -
exp(ar(v))
- Vr(v) +
Vr(v)
(
Theorem E.4 (Stationary Distribution). Assuming the solution space is bounded, as - 0c. the population distribution p" (v) converges to:
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1. Evolve: Converges the principal eigenfinction of the associated Schrödinger operator. Under the WKB approximation (o - O):
(- V2a f
PE (v) x exp
Vr(Vope) - r r(u)du)
(39)
Vopt
2. HELIX: Converges to a Boltamann-Gile Measure:
Pitetiz (v) x exp (2 (0))
(40)
Comment on theorem E.4
1. Concentration and Scaling. The concentration power of the stationary distributions deffiee as the inverse of their variance-exhilnn distinct scaling behaviors with respect to the noise pa- rameter Specifically, the concentration scales (1/or) for Evolve and ((1/a²) for HELIX. Given that a < in high-precision search contexts, it follows that 1/a² > 1/a. This inequality indicates that the sampling distribution of HELIX is exponentially optimum than that of Evolve. Under identical environmental conditions, HELIX achieves signifi- cantly more exhaustive exploration of the highly rewarded vieinity of the optimal solution.
concentrated around the
2. Intuitive Comparison (Quadratic Reward). To provide a concrete comparison, we analyze the behavior under local quadratic approximation of the reward function, r(v) Topt - & ||/l (centered at Vopt = 0). Deriving the exact Gaussian forms of the stationary distributions allows for a direct comparison of their variances, as summarized in Table 4
Table 4: Comparison Stationary Distributions under Quadratic Reward
Algorithm
Gaussian Form p" (v)
Variance Σ
Scaling vs.
nyk o2 ||o||?
a²
HELIX
x exp
2 Heliz
x o² (Sharper)
2np
Vak
Evolve
IX exp
||0:|2
2 Evo
X a (Broader)
2x
Vak
The ratio of their variances is given by:
Σ Hetliz
o2/2
Va 2nvk
o x
(41)
Evo a/Vak
o - 0. this ratio tends to zero. This rigorously confirms that HELIX's mechanism- -utilizini the gradient for directional wevementt-povvddee a superior capacity for stabilizing and concentrating the population compared to Evolve's reliance on scalar selection alone.
Potential for Further Reinforcement. It is worth noting that the current analysis assumes a uniform selection weight for HELIX (a 0). If we were to incorporate non-trivial selection weight w(v) = exp(ar(v)) into the HELIX framework, the final stationary distribution would theoretically become even more concentrated. Although a quantitative closed-form solution for process is mathematically intractable, qualitative analysis
this combined Drift-Diffusion-Selende suggests that this would further reinforce HELIX's focus and exploitation capabilities within high- reward regions.
F.3.2 PROOF OF THEOREM
Part I: HELIX -iffft-Diffusion. The dynamics follow the Langevin Equation dv, v/Vrrv;)dt+
adW, i. The probability density p(v,t) 1) evolves via the Fokker-Plant Planck equation:
Öp = - (p.pp.nn vr(vv) V2p.
(42)
Øt
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At steady state (0p/Ot 0), the probability flux J vanishes:
0² Vp' = 0 J = np<'vvr(v)
Vp-
21 Vr(v).
(43)
P
Integrating both sides yields Inp" r C, confirming the Boltzmann distribution.
Part II: Evolve Delection-Difffsionn The discrete selection-mutation process converges to the Replicator- Equation continuous time:
Op 0² - - p.
(44)
-
The stationary distribution p' satisfies the Schrödinger-like equation (where == =.)::
V' ar(()p" Ey".
(45)
2
Using the WKB Ansatz (v) = C(v) (epp(-Su)//5 and substituting into the equation, the lead- ing order terms (a 0) yield the Hamilton-Jacobi equation:
V/S22 ar(e) = E.
(46)
2
Setting the ground state condition at Uppt gives E = cr((vpp)) Solving for VS:
|VS( (ra((())pt) r(u)).
(47)
Integrating along the path from Uopt gives the action S(v), yielding the final asymptotic form for PF
G FORMALIZED ALGORITHM
In this appendix, we provide the detailed procedural description of the HELIX framework. Al- gorithm summarizes the full workflow, including sampling, prompt construction, model rollout, reinforcement learning updates, diversity estimation, and evolutionary population selection. These details complement the main text and offer complete specification of the method.
H EXAMPLE OF MODEL OUTPUT
We present examples of the best solutions found by our framework across different task categories. These visualizations highlight how HELIX generates high-quality and interpretable outputs di- verse scientific domains.
H.1 PHYSICS SIMULATION TASKS
Acoustic demultiplexer. Figure 14 displays the acoustic pressure field of our best-performing demultiplexer, which achieves a reward 14.260.
Iron core torque optimization. The best iron core design is shown in Figure 15 where the mag- netic flux density norm reaches a reward of 1.045.
Beam design. Figure 16 illustraaes the Mises stress pattern of the best beam structure discov- ered, which achieves reward of 17.298.
Meta-material optimization. The temperature distributions of the optimized meta-material under two loading conditions are presented in Figure 17 yiclding a reward of 1.278.
Inductor design. The optimized inductor is visualized Figure [8] with a magnetic flux density norm field corresponding to reward of 9.609.
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Algorithm HELIX Framework
Require:
Problem description p; initial solution(s) so: batch size B; GRPO group size G: number of
samples in prompt n; transition function T; reward function It; feedback function Fü; embedding model E.
Initialize dataset D = {50)-
2: Initialize population Po Do-
3: Initialize policy model parameters 0.
4: for iteration f = 0, 1, 2.... do
5: Sample B solutions from Pt, obtaining {st,))
> Prompt Construction
6: for
B do
7: Retrieve n ancestral states of 8t.1 (st))
8: Construct prompt:
qi ConstructPromppt( (p) U ( R(st,)) F(
{f (8L,),
(s
9:
end for
10:
for
= 1 to B do
b Model Rollout and Evaluation
11:
for j 1 to G do
12:
Generate action dij ~ no(- 9:).
Obtain new solution st41, T(s,,, a1s).
14:
Evaluate reward Tr+1,5J = R(s
Record feedback fi+1..) ((t+1ii.)
16:
end for end for
17:
18:
for
= to B do for j 1 to G do
> Reinforcement Learning Update
19:
20:
Compute normalized advantage:
Pet1ss T441. mean,fruti.s)
(td,(re+1ss))
end for end for
22:
23: Update policy: - 0 7 VoLGRPO.
24:
for = to B do
> Diversity Estimation
25:
for j 1 to G do
26:
Compute embedding 1441, = F((441.6))
Compute diversity score Div(set1As) (as Eq. (6)).
28:
end for
end for
30:
Update dataset: D,+1 + D. U (8:41.33)
Population Update
31: Use NSGA-II to select next population:
Pt+ Select ToPesATOPNSGA.I
D. 05.5141
U
32: end for
H.2
CIRCLE PACKING TASKS
Packing in square. As shown in Figure 19 our framework successfully packs 26 circles inside a square, achieving of radii of 2.6 2.6359830849 and surpassing the previous world record.
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0.1
1.5
0.08
0.06
1
0.04
0.5
0.02
0
-0.02
-0.04
-0.5
-0.06
-0.08
00.1
-1.5
-0.12
-0.1
-0.05
0.05
0.1m
Acoustic pressure distribution in the optimal acoustic demultiplexer design obtained by our framework, with reward 14.260.
Figure 14:
Packing in disk. Figure 20 demonstrates the packing 26 circles inside disk, reaching a total radius sum of 4.664465.
H.3 MACHINE LEARNING TASKS
We further demonstrate how our framework can be applied to classical machine learning problems, using both classification and regression benchmarks. The first example focuses on the Adult dataset, where we design rich set of engineered features that combine polynomial transformations, ratios, interaction terms, and domain-specific indicators. This structured feature space, coupled with LightGBM classifier and hyperparameter tuning, enables model to achieve a strong performance of 82.07 in macro Fl score (Figure 21).
For regression, turn to the Transparent Conductor Task. Here, we integrate multi-scale geometric deseriptors with spatial informatics derived from 3D Kernel Density Estimation and Principal Com- ponent Analysis. This framework transforms raw atomic coordinates into comprehensive feature set capturing both local coordination environments and global structural topology. With these en- hancements, model coupled with XGBoost regressor attains the second-highest score on the participants' leaderboard, corresponding to an RMSLE of 0,04915 (Figure 22p.
I
LLM USAGE STATEMENT
Large Language Models (LLMs) were used solely to aid writing and polishing the manuscript. All research ideas, experiments, and analyses were conceived and conducted by the authors, who take full responsibility for the content.
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0.05
1.4
0.04
0.03
1.2
0.02
0.01
0.8
0
-0.01
0.6
-0.02
0.4
-0.03
-0.04
0.2
-0.05
-0.04
-0.02
0
0.02
0.04
Figure 15: Magnetic flux density norm field for the optimized iron core configuration identified by our framework, achieving reward 11.045.
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x10
0.2
12
0.15
10
0.1
0.05
8
0
6
-0.05
-0.1
-0.15
2
-0.2
-0.2
-0.1
0
0.1
0.2m m
Figure 16: Von Mises stress distribution of the optimized beam design obtained by our framework, with reward 17.298.
X1U
294
xix
(a) Load
Load
Figure 17: Optimized temperature fields of the meta-materin designed by our framework under two different load conditions, achieving reward 1.278.
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0.45
0.4
0.35
2
0.25
0.2
0.15
0.1
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0.05
0
-0.05
-0.1
1
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-0.5
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-0.45
-0.2
0.2
0.4
0.6
Magnetic flux density norm field of the best inductor configuration identified by our framework, achieving reward 9.609.
Figure 18:
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Circle Packing (n=26, sum==.633983)
1.0
12
11
10
23
25
9
14
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7
19
24
0.2
20
16
17
18
22
0.0
0.0
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0.4
0.6
0.8
1.0
Figure 19: Arrangement of 26 circles within a square obtained by our framework, achieving pecord-breaking sum of radii of 2. 2.6398800..
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Circle Packing Unit Circle (n=26, sum= 46644655)
1.00
0.75
11
0.50
14
0.25
0.00
16
-0.25
22
-0.75
-1.00
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Figure 20: Optimized circle packing of 26 disks within a unit disk by our model, yielding a sum of radii of 4.664465.
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def engineer_feeeee featrrreeeuureeeeeeeee (X)
features = x. x.copy() ()
num_cols = [col for col in X. celumns if
!= 'object' and
col not in ('fnlwgt', "education-num")
# Core interaction features
features[ ('age__hp__proucct' ( week'l features| ccapptal_totaall capital-loss')
- features ['auuressaage ['age'] features [ hhours-pee
features! 'capital-gaia') features."
features (aatures(1oo__fnlwgtt np. (features [ ffllgt') + 1)
Enhanced polynomial features
for col in ['age', 'hours-per-wekk 'capital-gain'
'capital-loss' '1
(f [
[eatures
12
features **
13
# Age-based features with log transformation 1)
14
featurcs ('log_age1o = - [fcatures['ago)
Capital features with log transformations features 'capital_gain' np Lp. (arpruurreeeaaptallggaiaa. ('capital-gain') 1)
16
17
(. ('capitllllssss ( # Binned features for age and hours per week features ('a.ures(aag_grroupp)
features ''capital_loss'
18
cu. (featuurs [ ['age'l, bins-5, labels=Fal s=False)
features 'hour_group" labels=faise)
pd.cut (features) hours-per-week I, bins=5,
21
# Economic status features combining multiple variables features saaireesecccoomc_stttttiee
22
(features ['age'] / features ('hoursseehhhhpppee "hours-per- week'l) ["eaturees"capital__ooo
('agte"
23
Additional indicators for capital gains and losses
24
features| [
[has_capgain'] (features ''aapitllggin'] 0) .astype (int) (int)
25
features [hhas_caploss'] = (features ('capital-lose
0) .astype (int)
# Professional education indicator
((features (esaauurss"educatiiiii ['education'] 'Prof-
specialty')
(esaturre(""ducatiiii
'Exec-manageriall)
features 'educcliiocctiooon == 'Assoc-acdm' )) .astype
28
# Managerial education indicator
((features ['education'] -- "exec-
managerial')
centuurs(('duccttionn
'Assoc-voc') ) asstye
30
Interaction between numerical features interaction_com cols = l'age', "hours-per-week', 'capital_gaing gain', capital_loss
31
for i in range (interaction cols)) for j range (i+1, len (nteeractioonccol))))
33
34
coll interaction_colsti. co12 = interaction_cols] features [f [coll]]____[ccc]]]
35
fealures[coll] features [co12
37
# Ratio and difference features = features = ccapital / features
38
features I
'capital Loss' reppacee (U, 1)
Lositaal__sss""_ features capital_diift
features 'capital gain') features
capital_losst
40
41
return features
Figure 21: Python code feature engineering for solving classification task on Adult dataset. To- gether with LGBMClassifier and parameter search, our model achieved 82.07 marco fl score.
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def extract features (rawdata)
"""Returns pd.DataFrame of enhanced materials features. ..*
n**
Returns a pd. pd. of enhanced materials features.
features []
for rew rawdata iterrcws itwaraa.teerross/ ()
coords row row ["coordss"]]]
elements = rowl"elemetts"]
lattice_ rowl baaticc_eectoors.
11
Eificient atomic composition count using 1ist cemprehensions atom_counts ("Al": sum for el in elements if el 'A1 ))
12
13
"Ga": sum for el in elements if el -- "Ga
"in": sum for el in elements if el 'In'),
"0": (el sum for el in elements if in ('Al', 'Ga', 'In'))
16
# Structural features
struct _alculatee cocrdination, i _comrdinttiioo_mumerrr numberrrrneeeeeeemmooooooooornne (coords, elements)
mean_coords coords mean (axis=0)
19
centroid [mean_coords[0), mean_coords [1), mean_coords [2]]
20
Lattice vector features
21
a, b, с lattice_vec
vector_lengths [np. 1 nalllllll malg.norm((a) np. 1 minall.nnnn (b), np. ninlinalg
.norm [c)]
23
Volume calculation
24
volume abs (np.dot dot (np. cup.cros( (b, c), a))
# Angle calculations in radians
dot_ab np.dot (a, b)
27
det_ac np.dot (a, c)
28
dot_bc np. (b, c)
29
len_a = np.linalg Liinalg.norm(a
len_b np.linalg. mllinalg.normfff.... norm (b)
len_c np.linalg norm (c)
32
angle_ab ab np.arccos (dot_ab / (len_a len if (len_a len
!= else 0
33
angle_ac = np.arccos (dot_ac / (len_a len_e)) if (len_a len_c)
1- 0 else 0
angle_bc np.arccos (dot_bc (len_b len_c)) if (len_b len_c)
else
35
# Advanced structural features bond feats www.lllaeeeeaaaeeeee bend__ed______ Lengths (coords, elements)
36
Add and 3D kernel density estimates for atoric distribution from sklearn. sklearn.neighbors import KernelDensity from sklearn decomposition import PCA i Perform PCA to reduce dimensionality for density istimation pca = PCA (n_component==2) ccords_pca pea. www._ftaasffrrmcccoooo (coords)
38
39
40
43
# Calculate 2D kernel density
44
kde_ 2d KerneeDensity (bandiidttiidid fit (coords_po) censities_2d = ex.exp
45
(coords_pca))
Calculate 3D kernel density
kde_3d Kerne Density (bandwidth- fit (coords) densities_Be np. vd. sampl les (ccords))
48
49
Ceatures.appendl... $ All features listed above feature_ df pd. DataFraae (features) return feature df
52
Figure 22: Key pre-processing steps and geometric feature engineering implemented on the Trans- parent Conductor dataset. Together with an XGBRegressor, model achieved a score of 0.04915 in RMSLE, securing the second-highed position the participants' leaderboard.
71


Reviews:
[{"review_id": "KQIoG1ck4g", "reviewer": "Reviewer_c9MA", "summary": "This paper introduces HELIX, a hybrid method that enhances the reasoning capabilities of LLMs by combining reinforcement learning (RL) with verifiable rewards, evolutionary algorithms, and in-context learning. The authors argue that the RL fine-tuning (GRPO) phase increases domain-agnostic adaptation, the evolutionary nature promotes the solution diversity, and the in-context learning improves the solution quality using previous diverse high-quality solutions. This method is validated on several scientific and engineering tasks, showing improvement over baselines.", "questions": "1. Line 158, Eq. (2): does this objective guarantee Eq. (1)?\n2. Line 158, Eq. (2): why you take average over $s_t\\sim \\mathcal{P}$, not $\\sim \\mathcal{D}$? \n3. Figure 2: What is the lineage tree? An explanation seems to be needed for it.\n4. Line 211, Eq. (4): a) What is the formal definition of $q$? b) What is the formal definition of $\\hat{A}$? c) Can you define the MDP for your GRPO fine-tuning?\n5. Line 245, Eq. (5): a) Is this diversity measure new to this work, or borrowed from other works? b) It seems to require the pairwise distance calculation. Isn't it computationally heavy?\n6. Is the LLM finetuned to be used for different kinds of tasks? For example, for the Machine Learning benchmark, is a single LLM used for all the subtasks (Adult Income, Bank Marketing, and Boston Housing), or do we need to finetune an LLM for each subtask? \n7. How to generate the initial solutions?\n8. Doesn't HELIX suffer from a similar limitation as the evolutionary-algorithm-based approaches, designing problem-specific prompts for each task? If I understand correctly, HELIX needs prompts to combine previous solutions to generate a new one, just like the EAs need prompts to mix (crossover/mutation) solutions. How did you compare the complexity of designing the prompt between your algorithm and EAs?\n9. Line 253, what does \"nondominated sorting procedure\" mean?\n10. Line 256, what do \"a crowding distance measure\" and the \"objective space\" mean?\n11. Is \"Evolutionary Search\" in the title of section 3.2 indicating the in-context learning with previous solutions (\"ancestral states\" in Line 220)?\n\n---\n\n### LLM usage disclosure\nI did not use an LLM for my review, but I used grammar check software.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "1. This work effectively combines the known strategies for reasoning with LLMs, the RL fine-tuning, evolutionary algorithms, and in-context learning. To my knowledge, this combination is new.\n2. The proposed method shows strong empirical results compared to baselines across many scientific problems, showing its practicality.\n3. The source code is provided.", "weaknesses": "1. (Clarity) One of my main concerns is the clarity of the paper. While this work is understandable and easy-to-read at a high level, there is some ambiguity at low levels and many details seem to be missing. I left many questions to clarify some points that I couldn't understand; see the questions below. Additionally, I believe providing a detailed formalised algorithm and LLM prompts used for each experiment would be valuable. \n2. (Method) I believe GRPO is an on-policy RL algorithm, but the algorithm seems to train it with off-policy samples from the previous steps (since $(q,a) \\sim \\mathcal{D}$ where $\\mathcal{D}$ is previously discovered solutions). Please provide any rationale for this choice, or correct me if I'm wrong.\n3. (Evaluation) Some necessary evaluation settings seem to be missing, especially the standard deviation or confidence interval for each experiment. I assume that the proposed method consumes significantly more LLM calls than the current baselines (direct prompt, open evolve, task-specific methods), as it requires fine-tuning. Therefore, I believe that the fine-tuning method, e.g., vanilla GRPO, should also be included as a main baseline for Table 1.\n\n(Minor)  \n4. Line 90: Please add the reference for GRPO  \n5. Line 92: Using \"NSGA-II\" without any explanation makes readers confused; a short explanation of it (at least that it is an evolutionary algorithm) would be helpful.  \n6. Line 115-117: The related works are only roughly introduced. Providing more details regarding them would be valuable, e.g., how KL-Conv addressed the limitation, what the memory-less RL is, and why it cannot leverage previous solutions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3AEEJTfiwe", "reviewer": "Reviewer_Zrt5", "summary": "This paper introduces HELIX, a hierarchical evolutionary reinforcement learning framework that combines GRPO-based policy optimization, multi-objective evolutionary search (via NSGA-II), and in-context learning for solving open-ended scientific problems. The authors argue that HELIX enables LLMs to balance exploration and exploitation while iteratively improving solution quality. The method is tested on 19 diverse tasks (physics simulations, machine learning, symbolic regression, circle packing, and mathematical optimization) and achieves superior results on 16 tasks compared to strong baselines, including GPT-4o.", "questions": "- How sensitive is HELIX to the reward normalization constants and NSGA-II hyperparameters?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- Novel combination of reinforcement learning, evolutionary algorithms, and in-context prompting, addressing key limitations of existing approaches (e.g., entropy collapse, lack of diversity).\n\n- Strong empirical results across a wide range of domains, including physical design and scientific optimization, demonstrating the generality of the method.\n\n- Clear framework design and well-articulated motivation connecting HELIX’s components to the nature of open-ended scientific discovery.\n\n- Ablation and scaling analyses provide convincing evidence that both diversity-aware evolution and reinforcement learning contribute meaningfully to performance.", "weaknesses": "- While HELIX shows impressive performance across various scientific tasks, the paper provides limited discussion on the sensitivity of the framework to its key hyperparameters. In particular, the reward normalization constants (e.g., denominators used in Eq. 6–10 for physics tasks) and NSGA-II parameters (e.g., population size, crowding distance, KNN-based diversity metric) appear to play a crucial role in balancing exploration and exploitation, yet no analysis is offered regarding their influence on stability or convergence. This raises concerns about how robust HELIX remains under different hyperparameter settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "X4E4c1as5O", "reviewer": "Reviewer_RxUg", "summary": "The paper proposes HELIX, a framework to learn an LLM policy that iteratively produces diverse yet high quality solutions for open-ended problems.", "questions": "See weaknesses.", "limitations": "", "rating": 4, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper studies a setting of current interest to the community: open-ended problem solving.\n- The approach proposed in the paper, HELIX, is novel and interesting (though some details are unclear). \n- The experiments show the efficacy of the proposed method on interesting scientific reasoning tasks.", "weaknesses": "- My primary concern with the paper is with the description of the algorithm. It's unclear how NSGA-II is integrated into the pipeline. For instance, in Figure 2, the solutions selected via NSGA-II are passed to the RL algorithm as feedback. How does that work? \n- While the experiments are interesting, on many tasks, the Direct Prompt baseline works pretty well already. Additionally, on some tasks, Open Evolve performs worse than Direct Prompt. The authors should use more challenging domains to evaluate their method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "AjrngnRVkX", "reviewer": "Reviewer_TewV", "summary": "This paper proposes a unified framework that integrates reinforcement learning (via GRPO), evolutionary search (via NSGA-II), and in-context learning to enable large language models to autonomously tackle complex, unbounded scientific problems. HELIX maintains a diverse population of candidate solutions, embedding and filtering them for both reward and semantic diversity, while reinforcement learning updates the policy to progressively improve reasoning and exploration efficiency. Through iterative feedback loops, the model learns to build upon previous high-quality solutions, balancing exploration and exploitation. Evaluated on 19 tasks across five domains—including machine learning, physics simulation, circle packing, function minimization, and symbolic regression—HELIX consistently outperforms strong baselines, including GPT-4o and task-specific algorithms, and achieves state-of-the-art results such as a new world record in the circle-packing benchmark.", "questions": "- **\\[Q1]** The proposed framework mainly combines existing components—GRPO-style reinforcement learning, NSGA-II–based evolutionary selection, and in-context prompting. Could the authors clarify what specific technical or methodological challenges arise when integrating these elements, and how HELIX overcomes them? In particular, what makes this combination more than an engineering effort or straightforward composition of known techniques?\n\n- **\\[Q2]** The paper motivates HELIX as balancing exploration and exploitation through the interplay of RL, evolution, and in-context learning. However, no formal theoretical analysis or guarantees are provided. Could the authors clarify whether there is any theoretical grounding (e.g., convergence, stability, or optimality analysis) supporting this hybrid framework, or is it intended purely as an empirical system-level contribution?\n\n- **\\[Q3]** While HELIX shows consistent empirical improvements over baselines across multiple benchmarks, many of these tasks (e.g., UCI datasets, classical optimization functions, small-scale physics simulations) are relatively well-studied and limited in scale. Could the authors clarify how to interpret the significance of these improvements—do they demonstrate genuine scientific discovery capabilities, or primarily reflect engineering and optimization gains within benchmark settings? In other words, how do these results translate to more challenging, real-world, or open-ended scientific problems?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- **\\[S1] Important and timely problem.** The paper tackles the open-ended scientific problem-solving capability of LLMs, an increasingly important direction with potential impact across AI-driven discovery, optimization, and reasoning tasks.\n\n- **\\[S2] Thoughtful integration of multiple paradigms.** The proposed HELIX framework effectively combines reinforcement learning (GRPO-style policy optimization), evolutionary search (NSGA-II–based diversity–reward balancing), and in-context learning into a single hierarchical system. While each component is known, their coordination into an iterative, feedback-driven process is well-motivated and technically coherent.\n\n- **\\[S3] Comprehensive and diverse experimental evaluation.** The authors evaluate HELIX on 19 tasks spanning five domains—including machine learning, physics simulation, symbolic regression, and geometric optimization—demonstrating broad applicability and consistent improvements over strong baselines such as GPT-4o and AlphaEvolve-style systems.", "weaknesses": "See “Questions” below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)