PINN for solving PDE and Its application in Alzheimer’s disease

Overview

Its a research project focused on advancing the early diagnosis of Alzheimer’s disease by integrating partial differential equations (PDEs), Physics-Informed Neural Networks (PINNs), and symbolic regression. This project explores tau protein diffusion in the brain, aiming to predict the reaction term in the reaction-diffusion PDE that governs its behavior. Through this interdisciplinary approach, we aim to contribute to healthcare innovation by providing tools for better understanding Alzheimer’s progression and enhancing diagnostic precision.

The equation is given as:

$$\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot (D\cdot \mathrm{\nabla }c)+f(c)$$

Where:

• (c): Concentration
• (D): Diffusion coefficient (tensor or scalar)
• (f(c)): Reaction term

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Features

• 🧠 Tau Protein Diffusion Modeling: Simulated tau protein spread in the brain using the reaction-diffusion PDE.
• 🤖 Physics-Informed Neural Networks (PINNs): Combined physics and data to predict the reaction term in the PDE.
• 🔬 Symbolic Regression: Discovered interpretable mathematical expressions for the reaction term.
• 📊 Visualization: Simulated and visualized tau protein diffusion patterns in a 2D brain-like geometry.
• 🛠️ Web Interface: Dynamic webpage to interactively present diffusion results and project insights.
• 🧪 Data Integration: Used synthetic training data and PET scan data from the ADNI database for parameter extraction.

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How It Works

1. Training and Prediction

• Trained PINNs to predict the unknown reaction term in the reaction-diffusion PDE.
• Used symbolic regression to derive interpretable reaction equations.

2. Simulation

• Simulated tau protein diffusion in a 2D brain-like shape using COMSOL.
• Applied the predicted reaction term to visualize diffusion patterns.

3. Visualization

• Developed an interactive webpage to dynamically present results and insights.

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Getting Started

Prerequisites

• Python 3.8+
• COMSOL Multiphysics (for simulations)
• Jupyter Notebook

Installation

1. Clone the repository:

   git clone https://github.com/your-username/mind-code.git
   cd mind-code

2. Install dependencies:

   pip install -r requirements.txt

3. Run the result notebook:

   jupyter notebook

4. View the dynamic webpage:

   cd web_interface
   python -m http.server

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Results

Key Highlights

• Discovered reaction terms that match synthetic ground truth equations with high accuracy.
• Demonstrated diffusion patterns of tau protein in a 2D brain model.
• Improved PET scan quality using the heat equation for denoising.

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Future Work

• Extend the model to simulate 3D brain geometries.
• Test on additional real-world PET datasets.
• Explore the application of PINNs for other neurodegenerative diseases.

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Acknowledgments

• ADNI for providing PET scan data.
• COMSOL for simulation tools.

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