Large Language Models (LLMs) offer a promising avenue for scientific discovery, yet their application to symbolic regression is often constrained by inefficient search strategies and coarse feedback signals. Current methods typically guide LLMs using scalar metrics (e.g., global Mean Squared Error), which fail to identify which components of a proposed equation are driving performance or causing error. We introduce \textit{Influence-Guided Symbolic Regression} (IGSR), a method that frames equation discovery as an iterative two-step process combining diverse term generation with rigorous selection: an LLM generates candidate basis functions ψj(x) for a linear model, which are then evaluated using granular influence scores Δj. These scores quantify each term's marginal contribution to generalization accuracy, enabling an influence-guided pruning process that systematically refines the model structure. Integrating this mechanism into a Monte Carlo Tree Search (MCTS) enables navigating the combinatorial search space while balancing exploration of novel functional forms with exploitation of high-influence components. We demonstrate IGSR's effectiveness on a diverse suite of benchmarks, including LLM-SRBench, pharmacological PKPD models, an epidemiological simulation, and real-world genomic data. Notably, we validate the framework's capacity for genuine discovery in a case study using a high-dimensional biological dataset, in which IGSR identified a novel relationship between DNA methylation and RNA Polymerase II pausing; a hypothesis that was subsequently supported via wet-lab experimentation.
Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.
Symbolic regression aims to discover closed-form equations from data, but existing LLM-guided methods often rely on a unified proposal loop that compresses heterogeneous search failures into a scalar score and a single prompt. We propose A-SR, a self-evolving agentic framework that shifts the control unit from expression edits to role-conditioned evidence views. A-SR coordinates formula discovery through routing among coordination protocols, an online evaluator-reward role policy, and state-routed process memory. During search, evaluator feedback characterizes reliability and productivity, updates role-level utilities, and routes elite motifs, failure traces, and validity diagnostics to different agents. The framework self-evolves at two timescales: within a run, it adapts the search process without updating LLM parameters; across runs, recorded trajectories can be distilled into open-source LLMs as role-conditioned proposal priors. Averaged over the four LSR-Synth scientific domains in LLM-SRBench, A-SR improves Acc@0.01 over baselines from 25.79% to 48.30% with Llama3.1-8B, while A-SR-LoRA improves the corresponding Qwen3-4B result from 24.58% to 38.29%. On four real-world scientific discovery tasks, A-SR obtains the best in-distribution or out-of-distribution normalized mean squared error on 7 of 8 reported metrics.
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.