Know When the Equation Is Wrong: Structural Uncertainty for LLM-Guided Symbolic Regression under Extrapolation
TL;DR
LLM-SR-UQ is introduced, a framework that retains competing equation structures and uses their disagreement to characterize uncertainty about the underlying functional form and substantially improves coverage attainment by incorporating structural disagreement while preserving in-distribution coverage.
Abstract
Large language models (LLMs) can help discover mathematical equations from data, but most LLM-guided symbolic regression methods focus on a single best equation without quantifying uncertainty over alternative functional forms. This is challenging because distinct equations can fit the observed data similarly while producing substantially different extrapolations. Our experiments on the LLM-SRBench dataset show that standard conformal intervals achieve nominal in-distribution coverage but frequently fail under extrapolation. To tackle this challenge, we introduce LLM-SR-UQ, a framework that retains competing equation structures and uses their disagreement to characterize uncertainty about the underlying functional form. We extend the standard fixed-width conformal approach to quantify structural intervals, which measure disagreement among candidate equations and identify where prediction intervals should widen. Extensive experiments show our framework substantially improves coverage attainment by incorporating structural disagreement while preserving in-distribution coverage. Our approach also shows good generalizability when it is applied to other symbolic-regression methods.
