Influence-Guided Symbolic Regression: Scientific Discovery via LLM-Driven Equation Search with Granular Feedback

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

Scientists value simple, human-readable equations, because a clear formula reveals why something happens in a way that an opaque "black-box" AI cannot. Large language models (the technology behind chatbots) can suggest candidate equations, but they are usually guided only by a single overall "accuracy score". That score shows whether an equation is good, not which of its parts are doing the work, so the search is inefficient. We built a method, IGSR, that works differently. An LLM proposes the components of an equation; we then measure how much each components individually improves accuracy, keep those that are useful, drop the rest, and repeat. The result is a compact equation, in which, every term is meaningful and can be inspected. Across biological and medical datasets, IGSR discovers accurate, interpretable equations. When applied to real genomic data, it proposed a new biological hypothesis, that a chemical tag on DNA (methylation) suppresses stalling of the machinery that copies genes, which we then confirmed in the laboratory. This shows that AI can be a transparent, trustworthy partner in genuine scientific discovery.