2D Allen-Cahn Equation

Mathematical description

The PDE is given by

$\partial_tu-\lambda \Delta u+\epsilon u^3-\epsilon u=0,$

where $\lambda=0.0001$ and $\epsilon=1$ .

The domain is defined as

$(x,y,t) \in [0,1]^2 \times (0,1]$

The initial and boundary conditions are

$u(0,x,y)=u_0=\sum_{k=1}^{M}A_k\sin(2n_k^1\pi x+\phi_k^1)\sin(2n_k^2\pi y+\phi_k^2),\\ u(0,y,t)=u(1,y,t), \ u(x,0,t)=u(x,1,t), \ (y,t) \in [0,1] \times (0,1]\\ \partial_xu(0,y,t)=\partial_x(1,y,t), \ \partial_yu(x,0,t)=\partial_yu(x,1,t), \ (x,t) \in [0,1] \times (0,1]$

where $M=2, n_k^i \in \mathbb{Z}^+ \cap [1,8], A_k \in [0,1],\phi_k^i \in (0,2\pi), k=1,2 ,i=1, 2.$

Dataset file information

File name: 2D_Allen-Cahn_0.0001_1.hdf5

File hierarchy:

/
|-- tensor
|-- x-coordinate
|-- y-coordinate
|-- t-coordinate

Datasets information:

• tensor
• Shape: (1000, 101, 128, 128)
• Data Type: float
• Description: 1000 solution samples, each corresponding to a solution under one initial condition.
• x-coordinate
• Shape: (128,)
• Data Type: float
• Description: x-axis spatial coordinates
• y-coordinate
• Shape: (128,)
• Data Type: float
• Description: y-axis spatial coordinates
• t-coordinate
• Shape: (101,)
• Data Type: float
• Description: temporal coordinates

Data generation method

All samples are generated using spectral method based on Chebfun, an open-source package for numerical computing. Specifically, we use spin function to generate data.