Phase-Field AI Surrogate Solver

This repository implements a Convolutional Neural Network (U-Net) as an autoregressive surrogate solver for the Allen-Cahn equation. It demonstrates how Deep Learning can be used in the prediction of stiff nonlinear partial differential equations (PDEs) employed in materials science to model microstructural evolution.

The Governing Physics

The model learns and predicts the discrete time evolution of the Allen-Cahn equation. The phase-field order parameter $u$ evolves according to Allen-Cahn dynamics:

$$ \frac{\partial u}{\partial t} = L \left( \epsilon^2 \nabla^2 u - W(u^3 - u) \right) $$

Where:

• $u$: The non-conserved phase-field order parameter.
• $L$: Kinetic mobility coefficient.
• $\epsilon$: Gradient energy coefficient (controls interfacial energy and thickness).
• $W$: Double-well barrier height.
• $\nabla^2 u$: The spatial Laplacian, computed via a 2D 5-point stencil.

Project Architecture

The project is modularized for clean deployment:

• core/fdm_solver.py: A multi-core Finite Difference Method solver for ground truth Allen-Cahn dynamics.
• core/model.py: The U-Net PyTorch architecture designed for phase-field evolution.
• train.py: The data generation and training pipeline.
• evaluate.py: The inference script for autoregressive prediction on unseen random seeds.

Installation

Clone the repository and install the required dependencies:

git clone https://github.com/LingxiaS/PhaseField-CNN-solver
cd PhaseField-CNN-solver
pip install -r requirements.txt