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/advection.py

@@ -0,0 +1,107 @@
+"""Formulation of the advection problem."""
+
+import torch
+from ... import Condition
+from ...operator import grad
+from ...equation import Equation
+from ...domain import CartesianDomain
+from ...utils import check_consistency
+from ...problem import SpatialProblem, TimeDependentProblem
+
+
+class AdvectionEquation(Equation):
+    """
+    Implementation of the advection equation.
+    """
+
+    def __init__(self, c):
+        """
+        Initialize the advection equation.
+
+        :param c: The advection velocity parameter.
+        :type c: float | int
+        """
+        self.c = c
+        check_consistency(self.c, (float, int))
+
+        def equation(input_, output_):
+            """
+            Implementation of the advection equation.
+
+            :param LabelTensor input_: Input data of the problem.
+            :param LabelTensor output_: Output data of the problem.
+            :return: The residual of the advection equation.
+            :rtype: LabelTensor
+            """
+            u_x = grad(output_, input_, components=["u"], d=["x"])
+            u_t = grad(output_, input_, components=["u"], d=["t"])
+            return u_t + self.c * u_x
+
+        super().__init__(equation)
+
+
+def initial_condition(input_, output_):
+    """
+    Implementation of the initial condition.
+
+    :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
+    """
+    return output_ - torch.sin(input_.extract("x"))
+
+
+class AdvectionProblem(SpatialProblem, TimeDependentProblem):
+    r"""
+    Implementation of the advection problem in the spatial interval
+    :math:`[0, 2 \pi]` and temporal interval :math:`[0, 1]`.
+
+    .. seealso::
+
+        **Original reference**: Wang, Sifan, et al. *An expert's guide to
+        training physics-informed neural networks*.
+        arXiv preprint arXiv:2308.08468 (2023).
+        DOI: `arXiv:2308.08468  <https://arxiv.org/abs/2308.08468>`_.
+    """
+
+    output_variables = ["u"]
+    spatial_domain = CartesianDomain({"x": [0, 2 * torch.pi]})
+    temporal_domain = CartesianDomain({"t": [0, 1]})
+
+    domains = {
+        "D": CartesianDomain({"x": [0, 2 * torch.pi], "t": [0, 1]}),
+        "t0": CartesianDomain({"x": [0, 2 * torch.pi], "t": 0.0}),
+    }
+
+    conditions = {
+        "t0": Condition(domain="t0", equation=Equation(initial_condition)),
+    }
+
+    def __init__(self, c=1.0):
+        """
+        Initialize the advection problem.
+
+        :param c: The advection velocity parameter.
+        :type c: float | int
+        """
+        super().__init__()
+
+        self.c = c
+        check_consistency(self.c, (float, int))
+
+        self.conditions["D"] = Condition(
+            domain="D", equation=AdvectionEquation(self.c)
+        )
+
+    def solution(self, pts):
+        """
+        Implementation of the analytical solution of the advection problem.
+
+        :param LabelTensor pts: Points where the solution is evaluated.
+        :return: The analytical solution of the advection problem.
+        :rtype: LabelTensor
+        """
+        sol = torch.sin(pts.extract("x") - self.c * pts.extract("t"))
+        sol.labels = self.output_variables
+        return sol