1D Allen-Cahn Equation

1、Mathematical description

The PDE is given by

$\partial_tu-\lambda \partial_{xx}u+\epsilon u^3-\epsilon u=0,$

where $\lambda=0.0001$ and $\epsilon=5$ .

The domain is defined as

$(x,t) \in [-1,1] \times [0,1]$

The initial and boundary conditions are

$u(x,0)=u_0=\sum_{k=1}^{M}A_k\sin(n_k\pi+\phi_k),\\ u(-1,t)=u(1,t),\\ \partial_xu(-1,t)=\partial_xu(1,t).$

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

2、Dataset file information

File name: 1D_Allen_Cahn_0.0001_5.hdf5

File hierarchy:

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

Datasets information:

• tensor
• Shape: (10000, 101, 1024)
• Data Type: float
• Description: 10000 solution samples, each corresponding to a solution under one initial condition.
• x-coordinate
• Shape: (1024,)
• Data Type: float
• Description: spatial coordinates
• t-coordinate
• Shape: (101,)
• Data Type: float
• Description: temporal coordinates

3、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.