Inverse problem implementation (#177)

* inverse problem implementation

* add tutorial7 for inverse Poisson problem

* fix doc in equation, equation_interface, system_equation

---------

Co-authored-by: Dario Coscia <dariocoscia@dhcp-015.eduroam.sissa.it>

pina/problem/inverse_problem.py

@@ -0,0 +1,71 @@
+"""Module for the ParametricProblem class"""
+from abc import abstractmethod
+
+from .abstract_problem import AbstractProblem
+
+
+class InverseProblem(AbstractProblem):
+    """
+    The class for the definition of inverse problems, i.e., problems
+    with unknown parameters that have to be learned during the training process
+    from given data.
+
+    Here's an example of a spatial inverse ODE problem, i.e., a spatial
+    ODE problem with an unknown parameter `alpha` as coefficient of the
+    derivative term.
+
+    :Example:
+        >>> from pina.problem import SpatialProblem, InverseProblem
+        >>> from pina.operators import grad
+        >>> from pina.equation import ParametricEquation, FixedValue
+        >>> from pina import Condition
+        >>> from pina.geometry import CartesianDomain
+        >>> import torch
+        >>>
+        >>> class InverseODE(SpatialProblem, InverseProblem):
+        >>>
+        >>>     output_variables = ['u']
+        >>>     spatial_domain = CartesianDomain({'x': [0, 1]})
+        >>>     unknown_parameter_domain = CartesianDomain({'alpha': [1, 10]})
+        >>>
+        >>>     def ode_equation(input_, output_, params_):
+        >>>         u_x = grad(output_, input_, components=['u'], d=['x'])
+        >>>         u = output_.extract(['u'])
+        >>>         return params_.extract(['alpha']) * u_x - u
+        >>>
+        >>>     def solution_data(input_, output_):
+        >>>         x = input_.extract(['x'])
+        >>>         solution = torch.exp(x)
+        >>>         return output_ - solution
+        >>>
+        >>>     conditions = {
+        >>>         'x0': Condition(CartesianDomain({'x': 0}), FixedValue(1.0)),
+        >>>         'D': Condition(CartesianDomain({'x': [0, 1]}), ParametricEquation(ode_equation)),
+        >>>         'data': Condition(CartesianDomain({'x': [0, 1]}), Equation(solution_data))
+    """
+
+    @abstractmethod
+    def unknown_parameter_domain(self):
+        """
+        The parameters' domain of the problem.
+        """
+        pass
+
+    @property
+    def unknown_variables(self):
+        """
+        The parameters of the problem.
+        """
+        return self.unknown_parameter_domain.variables
+
+    @property
+    def unknown_parameters(self):
+        """
+        The parameters of the problem.
+        """
+        return self.__unknown_parameters
+
+    @unknown_parameters.setter
+    def unknown_parameters(self, value):
+        self.__unknown_parameters = value
+