Often, the role of a neural network is to minimize its parameters with respect to some given error criteria. This criteria, the cost or loss function, is a measure of how much error the output of the network has compared to some given known answers. A reformulation of (16) must therefore be done, such that it describes the problem a neural network can solve.
The neural network must find the set of weigths and biases \( P \) such that the trial solution in satisfies (16). The trial solution has been chosen such that it already solves the condition \( g(0) = g_0 \). What remains, is to find \( P \) such that $$ \begin{equation} g_t'(x, P) = - \gamma g_t(x, P) \tag{18} \end{equation} $$
is fulfilled as best as possible.