Enterprise AI Analysis
Knowledge integration for physics-informed symbolic regression using pre-trained large language models
This study demonstrates how pre-trained Large Language Models (LLMs) can be effectively integrated into Physics-informed Symbolic Regression (PiSR) to automate domain knowledge integration, enhance model robustness, and improve the reconstruction of physical dynamics.
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SR Algorithms Tested
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Pre-trained LLMs Used
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Physical Dynamics Evaluated
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Avg. SNR in Data
Executive Impact: Revolutionizing Scientific Discovery with AI
Our research introduces a novel framework that bridges the gap between traditional Symbolic Regression (SR) and the advanced reasoning capabilities of Large Language Models (LLMs). By incorporating an LLM-based scoring mechanism directly into the SR loss function, we've automated the integration of complex domain knowledge, traditionally a manual and expert-intensive task.
This integration significantly enhances the robustness of SR models against data noise and complexity, leading to more accurate and physically plausible equation discoveries. The framework was rigorously tested across diverse physical dynamics (dropping ball, simple harmonic motion, electromagnetic wave) using multiple SR algorithms (DEAP, gplearn, PySR) and leading LLMs (Falcon, Mistral, LLaMA 2).
The findings highlight a consistent improvement in the reconstruction quality of physical dynamics. Specifically, informative prompt engineering was shown to dramatically boost performance, suggesting that the quality of LLM guidance is a critical factor. This approach not only accelerates scientific discovery but also makes advanced symbolic regression accessible to a broader range of scientific problems by reducing the need for specialized human intervention.
Our method paves the way for a new era of AI-augmented scientific discovery, where intelligent systems can autonomously learn and validate physical laws from experimental data, accelerating research and innovation across disciplines.
Automated domain knowledge integration
Enhanced robustness to data noise
Improved equation plausibility and generality
Reduced manual feature engineering
Accelerated scientific discovery
Deep Analysis & Enterprise Applications
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Methodology Flowchart
Prompt Engineering Impact
Noise Robustness
Physical Dynamics & SR Algorithms
The proposed framework integrates LLMs into the Symbolic Regression process by adding an LLM-based evaluation term to the SR's loss function. This term evaluates the equation's dimensional consistency, simplicity, and physical realism based on the LLM's understanding of scientific literature. This creates a feedback loop that guides the SR algorithm towards physically plausible solutions.
Enterprise Process Flow
Function Governing Dynamics
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Generate Sample Dataset
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SR Model Proposes Equation (f_x)
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LLM Evaluates f_x (Dim. Corr., Simp., Realism)
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LLM Score Integrated into Loss Function
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SR Model Updates & Refines f_x
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Predicted Function (f_x')
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LLM Term in Loss Function
Prompt engineering significantly impacts performance. More informative prompts, especially those including variable descriptions and experiment context, lead to better results. Providing the Ground Truth (GT) equation further enhances the reconstruction quality, often leading to perfect structural recovery.
| LLM | SR | Prompt Variant | MAE | MSE | 1-R² | Exp. Tree Dist. |
|---|---|---|---|---|---|---|
| LLaMA | PySR | A (No Context) | 0.100 | 0.020 | 0.06 | 0.02 |
| Mistral | PySR | A (No Context) | 0.090 | 0.020 | 0.05 | 0.01 |
| LLaMA | PySR | B (Var. Desc.) | 0.090 | 0.018 | 0.05 | 0.01 |
| Mistral | PySR | B (Var. Desc.) | 0.080 | 0.018 | 0.04 | 0.01 |
| LLaMA | PySR | C (Exp. Desc.) | 0.080 | 0.016 | 0.04 | 0.00 |
| Mistral | PySR | C (Exp. Desc.) | 0.070 | 0.016 | 0.03 | 0.00 |
| LLaMA | PySR | D (Formula at End) | 0.040 | 0.005 | 0.02 | 0.00 |
| Mistral | PySR | D (Formula at End) | 0.030 | 0.004 | 0.01 | 0.00 |
| LLaMA | PySR | E (B+C) | 0.020 | 0.002 | 0.01 | 0.00 |
| Mistral | PySR | E (B+C) | 0.010 | 0.001 | 0.00 | 0.00 |
| LLaMA | PySR | H (B+C+D) | 0.020 | 0.002 | 0.01 | 0.00 |
| Mistral | PySR | H (B+C+D) | 0.010 | 0.001 | 0.00 | 0.00 |
LLM-integrated SR models demonstrate enhanced robustness to data noise, particularly under feature noise and combined noise scenarios. PySR consistently shows greater robustness across various noise levels and types.
| Noise Type | 1% | 2% | 3% | 4% | 5% |
|---|---|---|---|---|---|
| Feature noise | 0.11 | 0.15 | 0.20 | 0.22 | 0.25 |
| Target noise | 0.15 | 0.20 | 0.25 | 0.27 | 0.30 |
| Both noise | 0.25 | 0.30 | 0.35 | 0.37 | 0.40 |
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LLM-to-SR Runtime Ratio (Max)
The method was evaluated across three physical dynamics: dropping ball, simple harmonic motion, and electromagnetic wave, using DEAP, gplearn, and PySR. PySR generally outperforms other SR models in reconstruction quality, especially when guided by LLMs.
Free Fall Dynamics
For the 'free fall of a ball' scenario, the LLM-integrated SR consistently improved the discovery of the equation v = √2gh.
- Accurate reconstruction of square-root dependencies.
- Improved MAE and MSE metrics across all SR models when LLM guidance is used.
Simple Harmonic Motion
In the 'simple harmonic motion' case, the method successfully captured trigonometric functions, aiding in the discovery of x(t) = A cos(√(k/m)t + φ).
- Enhanced capture of periodic dynamics.
- Robustness to noise improved compared to baselines.
Mistral
Top Performing LLM
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