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Consider the following feed forward neural network with inputs $x_1$, $x_2$ and output $y$ and with inflowing weights $w_j$ ($j$ for each row of arrows) for the hidden layer and $u_j$ for the output layer and an activation function $h(x)$ and no bias neurons:

enter image description here

Can one mathematically model this as: $y=h( $$u_1$$h(w_1x_1+w_4x_2)$+$u_2$$h(w_2x_1+w_5x_2)$+$u_3$$h(w_3x_1+w_6x_2)$) and then use that expression after training the neural network for regression (plugging the example directly into the expression to find $y$)?

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Yes, that expression is right. It represents the feedforward propagation.

Just in case it helps, here is a way of writing that expression with matrix notation (it would be more worthy as the ANN size grows):

$$y = h\left[\begin{pmatrix} u_1 & u_2 & u_3 \end{pmatrix} h\left[ \begin{pmatrix} w_1 & w_4 \\ w_2 & w_5 \\ w_3 & w_6 \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \right] \right] $$

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    $\begingroup$ OP seems to say $u_i$ are the weights of output layer $\endgroup$
    – gunes
    Commented Sep 20, 2020 at 10:18
  • $\begingroup$ My bad, thanks for noticing, then I'm wrong and @Essam is correct. In a while I will delete my response to not confuse anyone. $\endgroup$
    – Javier TG
    Commented Sep 20, 2020 at 11:35
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    $\begingroup$ You can convert your answer into a positive one if you want. $\endgroup$
    – gunes
    Commented Sep 20, 2020 at 11:36

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