When Good Equations Get Bad Scores: Improving Symbolic Regression Through Better Parameter Optimization

Symbolic Regression (SR) plays a central role in scientific knowledge discovery by distilling mathematical equations from observational data. Most existing SR methods function within a bi-level optimization framework: an outer loop that searches for the discrete equation structure, and an inner loop that optimizes the continuous parameters of that structure. Crucially, parameter-fitting quality directly determines a structure's score and thus the outer-loop search. However, nonlinear operators make the inner loop highly non-convex, and budget-driven reliance on fast local solvers (e.g., BFGS) often yields poor local minima and underestimated scores for correct structures. This ``Good Structure, Bad Score'' phenomenon becomes a key bottleneck, degrading efficiency and misguiding the search away from the true equation. To resolve this, we propose SAGE-Fit (Structure-Aware and Semantics-Guided Evaluator for Symbolic Regression), an SR-native fitting framework that exploits the dual native priors of symbolic expressions. By capitalizing on the structural and semantic priors unique to SR, we design tailored modules for each property, thereby effectively mitigating this optimization bottleneck. Extensive experiments demonstrate that our approach, as a plug-and-play module, significantly enhances evaluation fidelity and universally improves the performance of various SR systems.

LLM-Based Scientific Equation Discovery via Physics-Informed Token-Regularized Policy Optimization

Symbolic regression aims to distill mathematical equations from observational data. Recent approaches have successfully leveraged Large Language Models (LLMs) to generate equation hypotheses, capitalizing on their vast pre-trained scientific priors. However, existing frameworks predominantly treat the LLM as a static generator, relying on prompt-level guidance to steer exploration. This paradigm fails to update the model's internal representations based on search feedback, often yielding physically inconsistent or mathematically redundant expressions. In this work, we propose PiT-PO (Physics-informed Token-regularized Policy Optimization), a unified framework that evolves the LLM into an adaptive generator via reinforcement learning. Central to PiT-PO is a dual-constraint mechanism that rigorously enforces hierarchical physical validity while simultaneously applying fine-grained, token-level penalties to suppress redundant structures. Consequently, PiT-PO aligns LLM to produce equations that are both scientifically consistent and structurally parsimonious. Empirically, PiT-PO achieves state-of-the-art performance on standard benchmarks and successfully discovers novel turbulence models for challenging fluid dynamics problems. We also demonstrate that PiT-PO empowers small-scale models to outperform closed-source giants, democratizing access to high-performance scientific discovery.