I am confused about the way the LSTM networks work when forecasting with a horizon that is not finite, but I'm rather searching for a prediction in whatever time in future. In physical terms, I would call it the evolution of the system.
Suppose I have a time series $y(t)$ (output) I want to forecast, and some external inputs $u_1(t), u_2(t),\cdots u_N(t)$ on which the series $y(t)$ depends.
It's common to use the lagged value of the output $y(t)$ as input for the network, such that I schematically have something like (let's consider for simplicity just lag 1 for the output and no lag for the external input):
$$ [y(t-1), u_1(t), u_2(t),\cdots u_N(t)] \to y(t) $$
In this way of thinking the network, when one wants to do recursive forecast it is forced to use the predicted value at the previous step as input for the next step. In this way we have an effect of propagation of error that makes the long term forecast badly behaving.
Now, my confusion is, I'm thinking of an RNN as a kind of a (simple version) implementation of a state-space model where I have the inputs, my output and one or more state variable responsible for the memory of the system. These variables are hidden and not observed.
So, now, the question: if there is this kind of variable taking already into account previous states of the system why would I need to use the lagged output value as input of my network/model?
Getting rid of this does my long term forecast would be better, since I'm not expecting anymore the propagation of the error of the forecasted output. (I guess there will be anyway an error in the internal state propagating)