fix tutorials latex and links (#261)
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Tutorial 7: Resolution of an inverse problem
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Tutorial: Resolution of an inverse problem
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============================================
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Introduction to the inverse problem
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@@ -7,26 +7,29 @@ Introduction to the inverse problem
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This tutorial shows how to solve an inverse Poisson problem with
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Physics-Informed Neural Networks. The problem definition is that of a
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Poisson problem with homogeneous boundary conditions and it reads:
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:raw-latex:`\begin{equation}
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\begin{cases}
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\Delta u = e^{-2(x-\mu_1)^2-2(y-\mu_2)^2} \text{ in } \Omega\, ,\\
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u = 0 \text{ on }\partial \Omega,\\
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u(\mu_1, \mu_2) = \text{ data}
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\end{cases}
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\end{equation}` where :math:`\Omega` is a square domain
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.. math::
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\begin{equation}
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\begin{cases}
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\Delta u = e^{-2(x-\mu_1)^2-2(y-\mu_2)^2} \text{ in } \Omega\, ,\\
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u = 0 \text{ on }\partial \Omega,\\
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u(\mu_1, \mu_2) = \text{ data}
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\end{cases}
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\end{equation}
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where :math:`\Omega` is a square domain
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:math:`[-2, 2] \times [-2, 2]`, and
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:math:`\partial \Omega=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4`
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is the union of the boundaries of the domain.
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This kind of problem, namely the “inverse problem”, has two main goals:
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- find the solution :math:`u` that satisfies the Poisson equation; -
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find the unknown parameters (:math:`\mu_1`, :math:`\mu_2`) that better
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fit some given data (third equation in the system above).
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* find the solution :math:`u` that satisfies the Poisson equation
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* find the unknown parameters (:math:`\mu_1`, :math:`\mu_2`) that better fit some given data (third equation in the system above).
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In order to achieve both the goals we will need to define an
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``InverseProblem`` in PINA.
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Let’s start with useful imports.
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``InverseProblem`` in PINA. Let’s start with useful imports.
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.. code:: ipython3
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