Source code for pina.problem.zoo.allen_cahn

"""Formulation of the Allen Cahn problem."""

import torch
from ... import Condition
from ...equation import Equation
from ...domain import CartesianDomain
from ...operator import grad, laplacian
from ...problem import SpatialProblem, TimeDependentProblem


def allen_cahn_equation(input_, output_):
    """
    Implementation of the Allen Cahn equation.

    :param LabelTensor input_: Input data of the problem.
    :param LabelTensor output_: Output data of the problem.
    :return: The residual of the Allen Cahn equation.
    :rtype: LabelTensor
    """
    u_t = grad(output_, input_, components=["u"], d=["t"])
    u_xx = laplacian(output_, input_, components=["u"], d=["x"])
    return u_t - 0.0001 * u_xx + 5 * output_**3 - 5 * output_


def initial_condition(input_, output_):
    """
    Definition of the initial condition of the Allen Cahn problem.

    :param LabelTensor input_: Input data of the problem.
    :param LabelTensor output_: Output data of the problem.
    :return: The residual of the initial condition.
    :rtype: LabelTensor
    """
    x = input_.extract("x")
    u_0 = x**2 * torch.cos(torch.pi * x)
    return output_ - u_0




class AllenCahnProblem(TimeDependentProblem, SpatialProblem):
    r"""
    Implementation of the Allen Cahn problem in the spatial interval
    :math:`[-1, 1]` and temporal interval :math:`[0, 1]`.

    .. seealso::
        **Original reference**: Sokratis J. Anagnostopoulos, Juan D. Toscano,
        Nikolaos Stergiopulos, and George E. Karniadakis.
        *Residual-based attention and connection to information
        bottleneck theory in PINNs*.
        Computer Methods in Applied Mechanics and Engineering 421 (2024): 116805
        DOI: `10.1016/
        j.cma.2024.116805 <https://doi.org/10.1016/j.cma.2024.116805>`_.

    :Example:
        >>> problem = AllenCahnProblem()
    """

    output_variables = ["u"]
    spatial_domain = CartesianDomain({"x": [-1, 1]})
    temporal_domain = CartesianDomain({"t": [0, 1]})

    domains = {
        "D": CartesianDomain({"x": [-1, 1], "t": [0, 1]}),
        "t0": CartesianDomain({"x": [-1, 1], "t": 0.0}),
    }

    conditions = {
        "D": Condition(domain="D", equation=Equation(allen_cahn_equation)),
        "t0": Condition(domain="t0", equation=Equation(initial_condition)),
    }