PINN-LB: Physics-Informed Neural Networks for the Leray-Burgers Equation

This repository contains the implementation of Physics-Informed Neural Networks (PINNs) for parameter estimation and adaptive solution of the Leray-Burgers equation, as presented in our research paper.

📄 Publication

Parameter estimation and adaptive solution of the Leray-Burgers equation using physics-informed neural networks

Authors: DooSeok Lee, Yuncherl Choi, Bong-Sik Kim

Journal: Results in Applied Mathematics, Volume 27 (2025), Article 100619

📖 Read the full paper

🔬 Abstract

This study presents a unified framework that integrates physics-informed neural networks (PINNs) to address both the inverse and forward problems of the one-dimensional Leray-Burgers equation. We investigate the inverse problem by empirically determining a physically consistent range of the characteristic wavelength parameter α, and solve the forward problem using a PINN architecture where α is dynamically optimized during training via our dedicated Alpha2Net subnetwork.

✨ Key Features

• Inverse Problem Solution: Systematic determination of physically consistent α parameter ranges
• Alpha2Net Architecture: Novel subnetwork for dynamic α optimization with physical constraints
• Unified Framework: Seamless integration of inverse and forward problem solutions
• Traffic State Estimation: Real-world application demonstrating practical utility
• Comprehensive Analysis: Comparison with viscous Burgers equation and convergence studies

🛠️ Requirements

System Dependencies

The code has been tested on Ubuntu 24.04.1 LTS. You must install TeX Live fonts outside of the Python environment:

sudo apt-get -qq install texlive-fonts-recommended texlive-fonts-extra cm-super dvipng

Python Environment

All required Python packages and dependencies are specified in the env-TF2N-ubuntu.yaml file. Create the conda environment using:

conda env create -f env-TF2N-ubuntu.yaml
conda activate pinn-lb

🚀 Quick Start

1. Clone the repository:

   git clone https://github.com/bkimo/PINN-LB.git
   cd PINN-LB

2. Set up the environment:

   # Install system dependencies
   sudo apt-get -qq install texlive-fonts-recommended texlive-fonts-extra cm-super dvipng
   
   # Create conda environment
   conda env create -f env-TF2N-ubuntu.yaml
   conda activate pinn-lb

3. Run the experiments:

   # Add specific commands to run your main experiments here
   python main_experiment.py

📁 Repository Structure

PINN-LB/
├── README.md
├── env-TF2N-ubuntu.yaml    # Environment configuration
├── src/                    # Source code
│   ├── inverse_problem/    # Inverse problem implementation
│   ├── forward_problem/    # Forward problem with Alpha2Net
│   └── traffic_estimation/ # Traffic state estimation application
├── experiments/            # Experimental scripts and notebooks
├── data/                   # Dataset files
└── results/                # Output results and figures

🎯 Main Contributions

1. Parameter Range Identification: Empirically determined α ranges (0.01-0.05 for continuous profiles, 0.01-0.03 for discontinuous profiles)

2. Alpha2Net Innovation: Novel subnetwork architecture that dynamically learns optimal α(t) while maintaining physical constraints

3. Unified Methodology: Seamless integration of inverse problem findings into forward problem solutions

4. Practical Application: Demonstrated efficiency in Traffic State Estimation with 2x speed improvement over traditional methods

🔗 Related Resources

• Paper URL: https://doi.org/10.1016/j.rinam.2025.100619
• Traffic Data: PISE Dataset
• Journal: Results in Applied Mathematics

📊 Results Highlights

• Successfully captures shock and rarefaction waves in the Leray-Burgers equation
• Achieves L₂ errors on the order of 10⁻² to 10⁻³ for various initial conditions
• Demonstrates superior computational efficiency in traffic state estimation applications
• Validates the physical consistency of learned α parameters across different scenarios

📝 Citation

If you use this code in your research, please cite our paper:

@article{kim2025parameter,
    title={Parameter estimation and adaptive solution of the Leray-Burgers equation using physics-informed neural networks},
    author={Lee, DooSeok and Choi, Yuncherl and Kim, Bong-Sik},
    journal={Results in Applied Mathematics},
    volume={27},
    year={2025},
    publisher={Elsevier},
    doi={10.1016/j.rinam.2025.100619}
}

👥 Authors

• DooSeok Lee - Daegu Gyeongbuk Institute of Science and Technology
• Yuncherl Choi - Kwangwoon University
• Bong-Sik Kim - American University of Ras Al Khaimah

🤝 Contributing

We welcome contributions! Please feel free to submit issues or pull requests.

📄 License

This project is licensed under the CC BY-NC License - see the paper's license terms for details.

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For questions about the implementation or paper, please open an issue or contact the corresponding authors.