🤖 AI

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

This paper introduces PiT-PO, a reinforcement learning framework that transforms Large Language Models into adaptive generators for symbolic regression by enforcing hierarchical physical validity and token-level structural parsimony, thereby achieving state-of-the-art equation discovery and enabling small-scale models to outperform closed-source giants.

Original authors: Boxiao Wang, Kai Li, Tianyi Liu, Chen Li, Junzhe Wang, Yifan Zhang, Jian Cheng

Published 2026-08-12

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Boxiao Wang, Kai Li, Tianyi Liu, Chen Li, Junzhe Wang, Yifan Zhang, Jian Cheng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but instead of looking for fingerprints or footprints, you are hunting for the secret mathematical recipe that governs how the world works. This field is called Symbolic Regression. While most computer programs today are like black boxes that guess an answer without explaining why, Symbolic Regression tries to find the actual equation—the "law of nature"—hidden inside a pile of data. Think of it like listening to a song and trying to write down the sheet music from scratch, rather than just humming along.

For a long time, scientists used old-school methods to find these recipes, but they were slow and often got lost in the vast wilderness of possible math formulas. Recently, a new kind of "super-brain" called a Large Language Model (LLM) has entered the game. These are the same types of AI that can write stories or chat with you, but they have also read millions of science textbooks. Because of this, they are incredibly good at guessing what a scientific equation might look like. However, there's a catch: these AI models are usually like a student who memorized the textbook but never took a test. They can spit out a formula that looks right and fits the numbers perfectly, but it might break the laws of physics or be unnecessarily complicated, like a recipe that calls for a cup of salt just because the chef felt like it.

This is where a new study comes in. The researchers, led by Boxiao Wang and Kai Li, wanted to fix this "memorizer" problem. They created a system called PiT-PO (Physics-informed Token-Regularized Policy Optimization). Instead of just asking the AI for a guess and hoping for the best, PiT-PO turns the AI into a student who actually learns from its mistakes in real-time. It's like having a strict but brilliant physics teacher standing right next to the AI, whispering, "No, that term doesn't make sense physically," or "You're using too many ingredients; simplify it!"

The team built a clever feedback loop. When the AI generates a candidate equation, PiT-PO checks it against two strict rules. First, it checks Physical Validity: Does this equation obey the laws of physics? For example, if the equation predicts that a fluid flows uphill without any energy, the system slaps a penalty on it. Second, it checks for Redundancy: Is the equation bloated with unnecessary parts? The researchers used a mathematical trick called the "Support Exclusion Theorem" to spot terms that are just noise and punish the AI for including them.

The results are impressive. In their tests, PiT-PO didn't just find equations that fit the data; it found the correct underlying laws, even when the data was messy. In one specific test involving the chaotic flow of air over hills (a problem known as "Flow over Periodic Hills"), the AI discovered a new way to model turbulence that was much closer to reality than traditional methods. Perhaps most excitingly, this system worked so well that even a small, open-source AI model (one that can run on a standard computer) was able to outperform massive, expensive "closed-source" giants.

By teaching the AI to learn from its own search process and strictly enforcing the rules of physics, PiT-PO turns a static guessing machine into an adaptive scientific discoverer. It suggests that we don't need to wait for super-computers to unlock the secrets of the universe; with the right training, smaller, more accessible AI tools can help us uncover the fundamental equations that drive everything from fluid dynamics to biological growth.

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

1. Problem Statement

Symbolic Regression (SR) aims to distill interpretable mathematical equations from observational data, serving as a cornerstone for uncovering physical laws. While recent approaches leverage Large Language Models (LLMs) to generate equation hypotheses by capitalizing on pre-trained scientific priors, existing frameworks suffer from two critical limitations:

Static Generation: Current methods treat LLMs as static generators, relying on prompt-level guidance (in-context learning) to steer evolutionary search. This paradigm fails to update the model's internal representations based on search feedback, preventing the model from internalizing evaluation signals or adapting its generation strategy to specific problem structures.
Physics-Agnostic Search: Existing frameworks often prioritize syntactic correctness over physical validity. Without rigorous constraints, LLMs frequently produce equations that fit data numerically but violate fundamental physical principles (e.g., dimensional homogeneity) or contain mathematically redundant structures, leading to overfitting and practical unusability.

The challenge is to transform the LLM from a static proposer into an adaptive generator that can dynamically align its generative distribution with the structural and physical characteristics of the target system.

2. Methodology: PiT-PO

The authors propose PiT-PO (Physics-informed Token-Regularized Policy Optimization), a unified framework that bridges LLM-driven evolutionary exploration with rigorous verification. The framework operates through a closed-loop evolutionary process driven by two synergistic mechanisms:

2.1 In-Search LLM Evolution

Unlike standard approaches that keep the LLM frozen, PiT-PO employs Group Relative Policy Optimization (GRPO) to update the LLM's parameters during the evolutionary search.

Mechanism: The LLM is treated as a policy $\pi_\theta$ generating token sequences. GRPO samples a group of outputs, evaluates them, and updates the policy parameters to maximize a surrogate loss.
Advantage: This "in-search" fine-tuning allows the model to consolidate effective symbolic patterns and guide subsequent exploration more efficiently, transforming general scientific priors into domain-specific expertise on the fly.

2.2 Dual-Constraint Learning Signals

To ensure the generated equations are both scientifically valid and structurally parsimonious, PiT-PO introduces a dual-constraint mechanism that generates fine-grained, token-level feedback.

A. Hierarchical Physical Constraints

A reward system enforces scientific validity through a hierarchical filter:

General-Level Constraints: Penalties for Dimensional Homogeneity ( $P_{dim}$ ) and Differentiability ( $P_{diff}$ ) to prune physically impossible structures (e.g., unit mismatches).
Domain-Specific Constraints: Expert knowledge is injected as inductive biases. For turbulence modeling, this includes constraints on Realizability (positive eigenvalues of Reynolds stress), Boundary Condition Consistency (stress decay at walls), Asymptotic Scaling (cubic scaling in viscous sublayers), and Energy Consistency.
Gated Activation: Physical penalties are activated only after a candidate equation reaches a baseline fitting accuracy threshold, allowing for "free" exploration in early stages before enforcing strict compliance.

B. Theorem-Guided Mathematical Constraints

To address mathematical redundancy, the authors introduce the Support Exclusion Theorem.

Theory: This theorem provides a theoretical guarantee to identify false discoveries (redundant terms) based on the magnitude of fitted coefficients relative to internal and external interference from other basis functions.
Token-Level Penalty: If a term is identified as redundant (its coefficient magnitude falls below a derived threshold), a token-level penalty ( $P_{tok}$ ) is applied to the specific tokens constituting that term. This penalty is logarithmic, imposing stronger penalties on terms with smaller coefficients.

2.3 Token-Aware Policy Update

Standard GRPO assigns a uniform advantage to all tokens in a sequence. PiT-PO refines this by synthesizing the global reward with the token-level penalty.

Token-Aware Advantage ( $\hat{A}_{i,k}$ ): The advantage for the $k$ -th token is calculated by standardizing the global reward and subtracting the local token penalty if the token belongs to a redundant term.
Effect: This creates a "dual-pressure" landscape where global rewards guide the policy toward physically consistent equations, while local penalties surgically excise redundant terms, ensuring the policy learns to suppress theoretically redundant structures explicitly.

2.4 Training Pipeline

The framework utilizes a multi-island topology to prevent premature convergence:

Island-Based Exploration: Multiple isolated islands evolve distinct lineages of equations to maintain search diversity.
In-Search Evolution: Trajectories from all islands are aggregated to perform policy optimization using LoRA (Low-Rank Adaptation) for efficiency.
Hierarchical Selection: A survival-of-the-fittest mechanism retains top-performing candidates and resets underperforming islands with high-fitness seeds.

3. Key Results

3.1 Benchmark Performance

PiT-PO was evaluated on the LLM-SR Suite and LLM-SRBench against state-of-the-art baselines (including GPlearn, PySR, uDSR, RAG-SR, and LLM-SR).

Accuracy: PiT-PO achieved state-of-the-art performance, recovering the highest number of ground-truth equations. On the LLM-SR Suite, it achieved 100% accuracy on Oscillation 1 and 99.99% on Oscillation 2 (using the 8B backbone). It significantly outperformed baselines on E. coli Growth and achieved competitive results on Stress-Strain, though it did not surpass the 4o-mini baseline on the latter.
Symbolic Accuracy: On LLM-SRBench (239 tasks), PiT-PO achieved the highest Symbolic Accuracy (SA) across all categories (Chemistry, Biology, Physics, Material Science), demonstrating superior ability to recover the correct symbolic form rather than just fitting numerically.
Efficiency: PiT-PO demonstrated faster convergence and the ability to escape stagnation regimes where baseline methods (like LLM-SR) plateaued at low-MSE but incorrect structures.

3.2 Empowerment of Small Models

A significant finding is that PiT-PO enables small-scale, open-source models to rival or outperform large, closed-source models in specific contexts.

Using a quantized Llama-3.2-1B model, PiT-PO achieved performance competitive with or superior to LLM-SR using Mixtral 8x7B and 4o-mini on Oscillation tasks and E. coli Growth. However, on the Stress-Strain task, the 1B model (76.91% accuracy) underperformed the 4o-mini baseline (85.33%), indicating that while the framework significantly bridges the capability gap, it does not universally surpass all proprietary giants across every metric.
This suggests that the in-search policy optimization effectively enhances smaller models, democratizing access to high-performance scientific discovery tools on consumer-grade hardware, particularly for tasks where structural discovery is paramount.

3.3 Real-World Application: Turbulence Modeling

The framework was applied to discover non-linear constitutive relations for Reynolds stress anisotropy in Flow over Periodic Hills.

Performance: The discovered model improved upon traditional RANS approaches (specifically the $k-\omega$ SST model) by producing anisotropic Reynolds stresses closer to Direct Numerical Simulation (DNS) references.
Physical Fidelity: The learned model showed enhanced physical consistency, reducing non-physical extremes and providing more accurate predictions of flow separation bubbles and skin friction coefficients compared to standard RANS and other SR-based methods.

3.4 Ablation Studies

Ablation studies confirmed the necessity of both components:

Removing physical constraints (w/o Phy) led to a substantial deterioration in NMSE and generalization.
Removing token regularization (w/o TokenReg) resulted in the persistence of redundant terms and a larger generalization gap between In-Distribution (ID) and Out-Of-Distribution (OOD) data.

4. Significance and Claims

The paper claims that PiT-PO fundamentally shifts the role of LLMs in scientific discovery from static proposers to adaptive, physics-aware generators.

Scientific Consistency: By integrating hierarchical physical constraints and theorem-guided token regularization, PiT-PO aligns generation with numerical fitness, scientific consistency, and parsimony simultaneously.
Democratization: The ability of PiT-PO to empower small models (e.g., 1B parameters) to rival closed-source giants on key discovery tasks establishes a practical methodology for automated scientific discovery that does not rely on massive compute resources or proprietary APIs, though performance varies by task complexity.
Practical Utility: The successful application to turbulence modeling demonstrates that the discovered symbolic corrections have tangible value in real-world engineering workflows, improving flow-field predictions beyond what is possible with standard linear models.

The authors conclude that this approach establishes a new state-of-the-art for SR benchmarks and offers a robust pathway for integrating domain-specific physical knowledge into the learning dynamics of LLMs.