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

2026-02-11

AI Summary

Existing large language models struggle to balance physical consistency and expression simplicity in symbolic regression and lack dynamic optimization mechanisms driven by search feedback. This work proposes the PiT-PO framework, which, for the first time, integrates physical constraints and token-level structural regularization into reinforcement learning–based policy optimization, enabling dynamic, hierarchical guidance of the equation generation process. The approach allows large language models to adaptively evolve into efficient scientific equation generators, achieving state-of-the-art performance on standard symbolic regression benchmarks. Notably, it successfully discovers a novel turbulence model and outperforms proprietary large models using only a small-scale architecture, significantly enhancing both the accuracy and interpretability of scientific discovery.

Technology Category

• Search and Optimization: Learning to Search
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language
• Machine Learning: Neuro-Symbolic Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• symbolic regression
• scientific equation discovery
• Large Language Models
• physical consistency
• mathematical redundancy

Innovation

• Symbolic Regression
• Large Language Models
• Reinforcement Learning
• Physics-Informed Constraints
• Token-Regularized Policy Optimization