Language models guide symbolic equation discovery by controlling search

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 $ψ_j(\mathbf{x})$ for a linear model, which are then evaluated using granular influence scores $Δ_j$. 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.

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.

Existing benchmarks for scientific equation discovery are largely composed of well-known equations available in the public domain, making it difficult to determine whether a model is discovering laws from data or merely recalling answers from its training corpus. LSR-Synth mitigates this problem by introducing novel synthetic terms into established scientific mechanisms and filtering the resulting tasks for novelty, solvability, and scientific plausibility. This paper examines a narrower measurement question: can these tasks further distinguish scientific priors supplied by language models from conventional operator search that does not access task semantics? We construct a semantics-free baseline using a fixed vocabulary with publicly documented provenance, and assess the role of candidate coverage through semantic blinding, library weakening, and matched operator-family knockouts. Under the current task snapshot, search budget, and scoring protocol, the fixed vocabulary already covers most tasks, while language-model-generated candidates rarely expand the set of solvable instances. Their marginal contribution becomes substantial only when vocabulary coverage is selectively disrupted. Strict out-of-distribution evaluation lowers the absolute success rates of all methods but does not alter this relationship. These findings neither invalidate LSR-Synth's controls against memorization of complete formulas nor imply that language-model priors are generally unhelpful. Rather, they support a more limited conclusion: most current tasks remain suitable for evaluating the fitting and recombination of previously unseen expressions, but are insufficient on their own to identify contributions from priors beyond a fixed search space.