fix tutorials latex and links (#261)

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Dario Coscia
2024-03-10 10:23:35 +01:00
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@@ -1,4 +1,4 @@
Tutorial 7: Resolution of an inverse problem
Tutorial: Resolution of an inverse problem
============================================
Introduction to the inverse problem
@@ -7,26 +7,29 @@ Introduction to the inverse problem
This tutorial shows how to solve an inverse Poisson problem with
Physics-Informed Neural Networks. The problem definition is that of a
Poisson problem with homogeneous boundary conditions and it reads:
:raw-latex:`\begin{equation}
\begin{cases}
\Delta u = e^{-2(x-\mu_1)^2-2(y-\mu_2)^2} \text{ in } \Omega\, ,\\
u = 0 \text{ on }\partial \Omega,\\
u(\mu_1, \mu_2) = \text{ data}
\end{cases}
\end{equation}` where :math:`\Omega` is a square domain
.. math::
\begin{equation}
\begin{cases}
\Delta u = e^{-2(x-\mu_1)^2-2(y-\mu_2)^2} \text{ in } \Omega\, ,\\
u = 0 \text{ on }\partial \Omega,\\
u(\mu_1, \mu_2) = \text{ data}
\end{cases}
\end{equation}
where :math:`\Omega` is a square domain
:math:`[-2, 2] \times [-2, 2]`, and
:math:`\partial \Omega=\Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4`
is the union of the boundaries of the domain.
This kind of problem, namely the “inverse problem”, has two main goals:
- find the solution :math:`u` that satisfies the Poisson equation; -
find the unknown parameters (:math:`\mu_1`, :math:`\mu_2`) that better
fit some given data (third equation in the system above).
* find the solution :math:`u` that satisfies the Poisson equation
* find the unknown parameters (:math:`\mu_1`, :math:`\mu_2`) that better fit some given data (third equation in the system above).
In order to achieve both the goals we will need to define an
``InverseProblem`` in PINA.
Lets start with useful imports.
``InverseProblem`` in PINA. Lets start with useful imports.
.. code:: ipython3