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