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Definition D5956
Feed-forward neural network

Let $N_0, N_1, \ldots, N_M \in \{ 1, 2, 3, \ldots \}$ each be a D5094: Positive integer such that
 (i) $W^{(m)} \in \mathbb{R}^{N_m \times N_{m - 1}}$ is a D4571: Real matrix for each $m \in \{ 1, \ldots, M \}$ (ii) $b^{(m)} \in \mathbb{R}^{N_m \times 1}$ is a D5200: Real column matrix for each $m \in \{ 1, \ldots, M \}$ (iii) $\phi^{(m)} : \mathbb{R}^{N_m \times 1} \to \mathbb{R}^{N_m \times 1}$ is a D5657: Real column matrix function for each $m \in \{ 1, \ldots, M \}$ (iv) $f_m : \mathbb{R}^{N_{m - 1} \times 1} \to \mathbb{R}^{N_m \times 1}$ is a D5657: Real column matrix function given by $f_m(x) = \phi^{(m)} \left( W^{(m)} x + b^{(m)} \right)$ for each $m \in \{ 1, \ldots, M \}$
The feedforward neural network with respect to $N_0$, $W^{(1)}, \ldots, W^{(M)}$, $b^{(1)}, \ldots, b^{(M)}$, and $\phi^{(1)}, \ldots, \phi^{(M)}$ is the D5657: Real column matrix function $$\begin{split} \mathbb{R}^{N_0 \times 1} \to \mathbb{R}^{N_M \times 1}, \quad x \mapsto f_M( f_{M - 1}( f_{M - 2}( \cdots f_2(f_1(x)) \cdots ) ) ) \end{split}$$