We can thus set up a recurring series which depends on the inputs \( x_i \) and \( F_i \) and the previous values \( h_{i-1} \). We assume now that the inputs at each step (or time \( t_i \)) is given by \( x_i \) only and we denote the outputs for \( \tilde{y}_i \) instead of \( v_{i_1} \), we have then the compact equation for our outputs at each step \( t_i \)
$$ y_{i}=h_i(x_i,h_{i-1}). $$We can think of this as an element in a recurrent network where our network (our model) produces an output \( y_i \) which is then compared with a target value through a given cost/loss function that we optimize. The target values at a given step \( t_i \) could be the results of a measurement or simply the analytical results of a differential equation.