Linking with RNNs

The equation

$$v_{i+1}=h_i(x_i,v_i,F_i).$$

can be used to train a feed-forward neural network with inputs $v_i$ and outputs $v_{i+1}$ at a time $t_i$. But we can think of this also as a recurrent neural network with inputs $v_i$, $x_i$ and $F_i$ at each time step $t_i$, and producing an output $v_{i+1}$.

Noting that

$$v_{i}=v_{i-1}+\Delta t\left(F_{i-1}-\eta v_{i-1}-x_{i-1}\right)=h_{i-1}.$$

we have

$$v_{i}=h_{i-1}(x_{i-1},v_{i-1},F_{i-1}),$$

and we can rewrite

$$v_{i+1}=h_i(x_i,h_{i-1},F_i).$$

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