Physics-Informed Neural Networks for Allen-Cahn Equations

This repository provides a simple and efficient implementation of Physics-Informed Neural Networks (PINNs) to solve the Allen-Cahn equations. PINNs leverage neural networks to directly solve partial differential equations (PDEs) by incorporating physical laws into the loss function.

Applications

Although the primary focus is on solving the Allen-Cahn equations, the techniques demonstrated here can also be applied to other similar equations, such as:

• Cahn-Hilliard equations: Used to model phase separation in binary mixtures and other phenomena involving free boundaries.
• Navier-Stokes equations: Governs fluid dynamics, modeling the behavior of incompressible flows.

Repository Structure

• AC.mat: Data file containing information for the Allen-Cahn equations.
• PINN_AC.ipynb: Jupyter Notebook with the full implementation of the PINN model, training, and solution visualization.
• save_figs: Directory containing saved figures from the experiments.

Requirements

To run this repository, ensure that the following libraries are installed:

• TensorFlow
• NumPy
• Matplotlib
• SciPy

You can install the necessary libraries using pip:

pip install tensorflow numpy matplotlib scipy