Adding new problems to problem.zoo (#484)

* adding problems
* add tests
* update doc + formatting

---------

Co-authored-by: Dario Coscia <dariocos99@gmail.com>

pina/problem/zoo/allen_cahn.py

@@ -0,0 +1,66 @@
+"""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>`_.
+    """
+
+    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)),
+    }