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This research advances symbolic regression by evolving Large Language Models into adaptive generators through reinforcement learning, ensuring generated equations adhere to physical laws while minimizing redundancy. The framework demonstrates that smaller models can outperform larger closed-source systems in discovering novel turbulence models for fluid dynamics problems. By integrating physics-informed constraints directly into the training policy, this work bridges the gap between statistical pattern matching and rigorous scientific discovery.
LLM-Based Scientific Equation Discovery via Physics-Informed Token-Regularized Policy Optimization
arXiv:2602.10576v2 Announce Type: replace-cross 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.
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.
Accepted at KDD 2026. Code is available at this https URL
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2602.10576 [cs.LG]
(or arXiv:2602.10576v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2602.10576 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 274 words
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