Documentation for v0.1 version (#199)

* Adding Equations, solving typos
* improve _code.rst
* the team rst and restuctore index.rst
* fixing errors

---------

Co-authored-by: Dario Coscia <dariocoscia@dhcp-015.eduroam.sissa.it>
This commit is contained in:
Dario Coscia
2023-11-08 14:39:00 +01:00
committed by Nicola Demo
parent 3f9305d475
commit 8b7b61b3bd
144 changed files with 2741 additions and 1766 deletions

View File

@@ -31,12 +31,16 @@ The problem definition
----------------------
The two-dimensional Poisson problem is mathematically written as:
:raw-latex:`\begin{equation}
\begin{cases}
\Delta u = \sin{(\pi x)} \sin{(\pi y)} \text{ in } D, \\
u = 0 \text{ on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
\end{cases}
\end{equation}` where :math:`D` is a square domain :math:`[0,1]^2`, and
.. math::
\begin{equation}
\begin{cases}
\Delta u = \sin{(\pi x)} \sin{(\pi y)} \text{ in } D, \\
u = 0 \text{ on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
\end{cases}
\end{equation}
where :math:`D` is a square domain :math:`[0,1]^2`, and
:math:`\Gamma_i`, with :math:`i=1,...,4`, are the boundaries of the
square.
@@ -158,9 +162,10 @@ is now defined, with an additional input variable, named extra-feature,
which coincides with the forcing term in the Laplace equation. The set
of input variables to the neural network is:
:raw-latex:`\begin{equation}
[x, y, k(x, y)], \text{ with } k(x, y)=\sin{(\pi x)}\sin{(\pi y)},
\end{equation}`
.. math::
\begin{equation}
[x, y, k(x, y)], \text{ with } k(x, y)=\sin{(\pi x)}\sin{(\pi y)},
\end{equation}
where :math:`x` and :math:`y` are the spatial coordinates and
:math:`k(x, y)` is the added feature.
@@ -249,9 +254,10 @@ Another way to exploit the extra features is the addition of learnable
parameter inside them. In this way, the added parameters are learned
during the training phase of the neural network. In this case, we use:
:raw-latex:`\begin{equation}
k(x, \mathbf{y}) = \beta \sin{(\alpha x)} \sin{(\alpha y)},
\end{equation}`
.. math::
\begin{equation}
k(x, \mathbf{y}) = \beta \sin{(\alpha x)} \sin{(\alpha y)},
\end{equation}
where :math:`\alpha` and :math:`\beta` are the abovementioned
parameters. Their implementation is quite trivial: by using the class