Organizations: Centre for Data Science and Artificial Intelligence & School of Engineering and Computer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand · Department of Computer Science and Engineering, Michigan State University, East Lansing, MI 48824, USA
Abstract
Parent selection significantly affects exploration, exploitation, and complexity control in genetic programming (GP) for symbolic regression. It is unclear whether large language models (LLMs) can synthesize effective operators in a zero-shot setting without iterative meta-evolution. Here, zero-shot means that the model receives only the task description, with no reference operators or iterative feedback. In this work, we benchmark zero-shot synthesis of parent-selection operators across eight LLMs within a standard GP framework for symbolic regression. Each model receives the same natural-language prompt to generate a parent-selection operator, which is then evaluated in a standard GP framework with only the parent-selection operator replaced, while all other components and the evolutionary-search budget are held constant. For each LLM, ten independent zero-shot operators are evaluated on twelve OpenML regression benchmarks and compared against automatic lexicase and tournament selection baselines. Claude Sonnet4.6 and Gemini3.1 Pro stand out for consistently strong performance on both training and held-out test R2. The strongest operator in our benchmark---a Kimi~K2.5 zero-shot synthesis---surpasses the automatic lexicase and tournament baselines in search effectiveness. These results suggest that zero-shot LLM synthesis is a viable approach to generating competitive GP selection operators. Analysis shows that many generated operators use semantics to guide selection, suggesting that LLMs can produce non-trivial search heuristics from the task description alone. We also examine the relationship between public LLM leaderboard rankings and GP performance. Widely used benchmarks, such as Humanity's Last Exam and SWE-bench Verified, strongly correlate with training R2, while their relationship to held-out test R2 is weaker and less clear.
Symbolic regression (SR) discovers compact mathematical expressions from data, yet recent LLM-based evolutionary methods remain sample-inefficient because they rely mainly on scalar feedback such as MSE. We identify a core limitation: existing methods conflate candidate proposal with search guidance, requiring the LLM to infer how to evolve an expression, diagnose its errors, and reuse past experience from a single score. To address this, we propose Deliberate Evolution (DE), an agentic framework that decouples symbolic generation from search control. DE guides LLM proposals with adaptive operators for search direction, analytical tools for structural diagnosis, and reflective memory for trajectory-level experience. Experiments on LLM-SRBench show that DE consistently outperforms representative LLM-based SR baselines across diverse scientific domains while using only 40% of the standard sample budget.
Symbolic regression aims to uncover explicit scientific laws from data. Recent methods use LLMs to guide mutation from background text, which is more directed than random genetic programming. However, exact symbolic recovery requires both semantic guidance and explicit structure, so that domain-informed search are carried out through valid symbolic representation. Current LLM-driven systems remain structure-blind: they select among opaque candidates, lack explicit mechanisms for local mutation, and rely on brittle coefficient fitting that can undervalue correct skeletons. We propose FunctionEvolve, an evolutionary framework using expression trees to organize the whole search: structural summaries promote diverse parent selection, local tree edits preserve useful subexpressions, and structure-aware fitting decomposes, constrains, and simplifies coefficients for more reliable scoring. It uses only elementary function families, without additional domain-specific rules limiting generalization. On the 129-task synthetic subset of LLM-SRBench, FunctionEvolve with \emph{Claude Opus 4.6} recovers 107 exact forms, reaching 82.9% SA@50, 4.5x above same-backbone baselines, and 55.8% SA@1, 3.6x above the strongest previously published top-1 result. Ablations show that structure-visible search is central to reliable recovery, with LLM-guided refinements and structure-aware coefficient optimization serving as essential proposal and scoring mechanisms. We also audit the benchmark and show that collinearity in its materials-science subset creates identifiability issues.
Genetic Programming and its variants, such as grammatical evolution, are widely used in Symbolic Regression to derive mathematical expressions from multivariate data. In addition to predictive accuracy, models are appreciated for their potential to provide interpretability, offering explicit equations that relate input variables to outcomes. However, achieving interpretability and plausibility remains challenging, as evolved models may be complex or scientifically inconsistent. In this study, we explore whether Large Language Models, can assist in improving the explainability of Symbolic Regression models generated by evolutionary computation methods. Building upon our previous work on estimating body fat percentage using grammar-based Genetic Programming , we investigate the use of LLMs as post-processing tools to analyze and rank evolved expressions according to their interpretability and medical plausibility. Four symbolic expressions are analysed by three LLMs over three repeated runs, and the resulting interpretations and rankings are assessed by a panel of three clinicians. Across the three LLMs, comparative model-ranking outputs received more favorable clinician assessments than isolated term-level interpretations. However, the LLMs also produced physiologically and mathematically questionable explanations, indicating that they are better suited to comparative auditing under expert oversight than to autonomous validation.\blfootnote{The present work is an extended version of a paper submitted into a journal.
Jorge López-Varela, J. Ignacio Hidalgo, José-Manuel Muñoz +6