Neural network

A neural network consists of the following information:

• A directed acyclic graph $N = (V,E)$ where the source vertices are called the input layer $I \subset V$ and the sink vertices are called the output layer $O \subset V$, and
• For each $v \in V \backslash I$ we have a perceptron $p_v: \mathbb{R}^{\vert In(v) \vert} \rightarrow\mathbb{R}$.

In the Modelling framework this would simulate something with $A \subset \mathbb{R}^{\vert I \vert}$ and $B \subset \mathbb{R}^{\vert O \vert}$. To run the model you compute a value for each vertex $x_v \in \mathbb{R}$ for $v \in V$. For input $(a_i)_{i \in I}$ we associate $a_i =: x_i$ to vertex $i \in I$. Then for all $y \in V \backslash I$ once all of $In(y)$ have values we set

$$x_y := p_y((x_v)_{v \in In(y)}).$$

To generate output vector $(x_o)_{o \in O} \in B$.

Normally the topology of $N$ is chosen so that the vertices $V$ can be partitioned into $L_i$ for some $1 \leq i \leq k$. Where for all $v \in L_i$ $In(v) = L_{i-1}$ and $Out(v) = L_{i+1}$. Where $L_1$ is the input layer and $L_k$ is the output layer all other layers are called hidden layers.