A-SR: Self-Evolving Agentic LLMs for Symbolic Regression via Hierarchical Coordination

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.

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.