For a small project I have been looking at applying primarily RNN architectures to a multivariate time series problem. However, research that provides reasonable assumptions on how to model the time series in terms of the residual and a neural network seems to be hard to find. Although for long term simulation of scenarios this seems to be a core aspect of analyzing time series, considering classical time series research.

I would be very grateful for pointing out good papers concerning neutral nets in time series or would like to talk about what kind of stochastic model people having experience chose and why. As a starting point I suggest RNN architectures can fit the trend, seasonality and an ARMA(p,q) model. However modelling time dependent variance and extremal values seems to be a rather hard task for neural nets.

Sorry for not having great academic english and looking forward to your answers!