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

TL;DR

This paper introduces PiT-PO, a reinforcement learning framework that fine-tunes LLMs to generate physically valid and concise scientific equations, improving discovery accuracy and efficiency in symbolic regression tasks.

Contribution

The work presents a novel physics-informed, token-regularized reinforcement learning method to adapt LLMs for more accurate and physically consistent equation discovery.

Findings

Achieves state-of-the-art results on benchmark symbolic regression tasks.

Successfully discovers new turbulence models in fluid dynamics.

Enables small models to outperform larger, closed-source models.

Abstract

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.