CoEvo: Continual Evolution of Symbolic Solutions Using Large Language Models

Conference: AAAI 2026 arXiv: 2412.18890 Code: Available Area: LLM/NLP Keywords: Symbolic Regression, LLM-based Evolutionary Search, Knowledge Library, Open-ended Innovation, Multi-representation Space

TL;DR

This paper proposes CoEvo, a framework that integrates LLMs with evolutionary search methodology to achieve continual open-ended evolution of symbolic solutions through a dynamic knowledge library and multi-representation spaces (natural language / mathematical formulas / code), significantly outperforming existing symbolic regression methods on the AI Feynman benchmark.

Background & Motivation

The discovery of symbolic solutions — mathematical expressions, logical rules, algorithmic structures — is foundational to scientific and engineering progress. However, existing approaches face two major bottlenecks:

Traditional methods (evolutionary algorithms such as PySR, deep learning methods such as NeSymReS): low search efficiency, difficulty integrating knowledge effectively.

LLM-based methods (FunSearch, LLM-SR): improved search efficiency, but lack the ability to continuously refine and extend discovered solutions and their underlying knowledge, limiting open-ended innovation.

Core problem: Can LLMs not only reuse existing knowledge but also discover new knowledge and evolve continuously?

CoEvo's vision: to define the discovery of symbolic solutions as a lifelong, iterative process — analogous to human scientific exploration — where solutions and foundational knowledge co-evolve.

Method

Overall Architecture

CoEvo consists of three core components:

CoEvo Framework
├── Idea Tree-based Solution Generation
│   ├── Step 1: Inspiring
│   ├── Step 2: Thinking
│   └── Step 3: Solving
├── Evolutionary Search
│   ├── Initialization
│   ├── Crossover (Positive / Negative)
│   ├── Mutation (Positive / Negative)
│   └── Population Update (Elitism)
└── Knowledge Library (Dynamic)
    ├── Summarization
    ├── Management (Clustering & Deduplication)
    └── Reuse (Random / Similarity-based)

Key Designs

1. Idea Tree-based Solution Generation

This component simulates a three-step human problem-solving process:

Step	Human Analogy	LLM Operation	Purpose
Inspiring	Obtaining initial inspiration	Retrieve relevant ideas from the knowledge library	Stimulate diversity
Thinking	In-depth reasoning	Iteratively refine ideas based on evaluator feedback	Improve quality
Solving	Output solution	Generate solutions in multiple formats	Explore multiple spaces

Tree structure: starting from \(N_0\) root ideas, each layer evolves \(N_k\) ideas based on the parent ideas and evaluator feedback. Unlike the exhaustive branching of Tree-of-Thought, a constrained network structure is adopted to avoid exponential computational overhead.

2. Multi-Representation Solutions

The search space is extended from the traditional mathematical formula/code representations to three levels:

Representation Space	Complexity	Knowledge Richness	Implementation
Mathematical Formula	Low	Low	LaTeX code
Python Code	Medium	Medium	Executable code
Natural Language	High	High	LLM reasoning text

Key insight: different representation spaces encode different levels of knowledge — the natural language space is the richest, enabling full exploitation of LLMs' reasoning capabilities.

3. Dynamic Knowledge Library

Three core mechanisms:

Summarization: - Trigger condition: a solution achieves a higher score during tree-based search or offspring generation - Operation: the LLM extracts and summarizes the key ideas behind the improvement, stored in a definition-description format - Purpose: learning "why a solution is good" from high-quality solutions

Management: - The library maintains a fixed capacity (30 entries in experiments) - Semantic clustering based on cosine similarity of sentence embeddings (DBSCAN) - Representative ideas are retained; redundant ones are removed

Reuse: two modes

Mode	Usage Scenario	Strategy
Random Reuse	Generating new solutions	Randomly sample one idea from each cluster
Similarity-based Reuse	Tree-based idea search	Retrieve ideas most similar to the current idea

4. Evolutionary Search

Operator	Type	Description
Crossover	Positive	Promotes solutions similar to parent ideas
Crossover	Negative	Promotes solutions divergent from parent ideas, enhancing diversity
Mutation	Positive	Small incremental modifications
Mutation	Negative	Large, significant changes
Population Update	Elitism	Retain the top \(N\) solutions by score

Loss & Training

• Evaluation metric: Normalized Mean Squared Error (NMSE), measured both in-distribution (ID) and out-of-distribution (OOD)
• Iteration budget: 2,000 iterations, 100 generations, 20 samples per generation
• Knowledge library capacity: 30 entries
• LLM backbone: gpt-3.5-turbo and gpt-4o-mini
• No gradient-based training — the method is entirely based on LLM generation and evolutionary search

Key Experimental Results

Main Results

Table 1: Performance Comparison on AI Feynman Benchmark (NMSE)

Method	Oscillation 1 ID/OOD	Oscillation 2 ID/OOD	E. coli Growth ID/OOD	Stress-Strain ID/OOD
GPlearn	0.0155/0.5567	0.7551/3.188	1.081/1.039	0.1063/0.4091
PySR	0.0009/0.3106	0.0002/0.0098	0.0376/1.014	0.0331/0.1304
uDSR	0.0003/0.0007	0.0032/0.0015	0.3322/5.458	0.0502/0.1761
LLM-SR (gpt-4o-mini)	5.14e-9/3e-4	1.79e-7/3.11e-5	0.0214/0.0264	0.0020/0.0020
CoEvo (gpt-3.5-turbo)	4.32e-9/8.71e-5	1.58e-10/1.32e-10	1.58e-9/1.21e-8	0.0020/0.0015

CoEvo outperforms LLM-SR on Oscillation 2 and E. coli Growth by several orders of magnitude.

Table 2: Comparison of Search Spaces Across Methods

Method	Search Space	Knowledge Management	Open-ended Evolution
PySR	Formula/Code	None	No
FunSearch	Code	None	No
LLM-SR	Code	Static	No
EoH	Natural Language + Code	None	No
CoEvo	Natural Language + Formula + Code	Dynamic Knowledge Library	Yes

Ablation Study

• Impact of the knowledge library: with vs. without the library, NMSE on E. coli Growth improves by 2–3 orders of magnitude.
• Impact of LLM choice: gpt-3.5-turbo and gpt-4o-mini yield comparable performance, indicating the method is not sensitive to the choice of backbone LLM.
• Proportion of valid solutions: CoEvo generates a significantly higher proportion of valid solutions than LLM-SR across all benchmarks.
• Cross-source knowledge experiment: knowledge extracted from gpt-3.5-turbo applied to gpt-4o-mini (and vice versa) consistently improves the quality of new solutions.

Key Findings

1. Discovery of implicit solutions for Oscillation 2: CoEvo is the only method to discover that numpy.gradient can be applied to velocity data to compute acceleration — a non-traditional, data-driven approach — while all other methods attempt to recover explicit physical equations.
2. Visualization of knowledge evolution: the knowledge library evolves dynamically during the search process; upon discovering the implicit solution, the diversity of the library expands rapidly.
3. Need for knowledge condensation: not all accumulated knowledge is useful; future work requires an idea condensation mechanism to filter out uninformative entries.
4. Valid solution proportion is a core advantage of CoEvo — a better exploration strategy reduces invalid sampling.

Highlights & Insights

• First work to define symbolic discovery as a lifelong, continual process: the goal is not only to find solutions but to continuously refine knowledge and expand discovery capabilities.
• Elegant multi-representation space design: the knowledge richness of the natural language space compensates for the limitations of traditional formula/code spaces.
• Closed-loop knowledge pipeline — collection, management, and reuse: summarization extracts knowledge from high-quality solutions, management prevents knowledge bloat, and reuse injects knowledge at the right moment.
• The Oscillation 2 case is a standout highlight: CoEvo discovers a non-traditional solution path that even human researchers might overlook.

Limitations & Future Work

1. Experiments cover only 4 AI Feynman problems; the scale is limited, and generalizability requires validation on more benchmarks.
2. The knowledge library capacity (30 entries) is set empirically, lacking theoretical justification.
3. Reliance on LLM API calls introduces cost and latency constraints that limit large-scale deployment.
4. Idea condensation is mentioned by the authors but not implemented; this may be a key avenue for further performance improvement.
5. No comparison is made with the latest code generation or scientific discovery LLMs (e.g., AlphaCode, AlphaGeometry).

Related Work & Insights

• FunSearch (Paredes et al. 2024): a pioneering LLM + evolutionary search framework, but restricted to code space with no knowledge management.
• LLM-SR (Shojaee et al. 2024): the current state-of-the-art symbolic regression method; CoEvo builds upon it by adding the knowledge library and multi-representation spaces.
• Tree-of-Thought (Yao et al. 2024): tree-structured reasoning; CoEvo's idea tree draws inspiration from this paradigm while avoiding exponential computational overhead.
• Implications for model compression: during knowledge distillation, the "knowledge" of a teacher model could similarly be dynamically summarized, managed, and refined, rather than being extracted in a one-shot, static manner.

Rating

• Novelty: ⭐⭐⭐⭐ (first to define symbolic discovery as a continual evolution process; knowledge library design is original)
• Experimental Thoroughness: ⭐⭐⭐ (benchmark scale is limited, but the analysis is in-depth)
• Writing Quality: ⭐⭐⭐⭐ (framework diagrams are clear; ablation study is comprehensive)
• Value: ⭐⭐⭐⭐ (opens a new paradigm for LLM-driven scientific discovery)

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