  In this paper, we consider high-dimensional stationary processes where a new
observation is generated from a compressed version of past observations. The
specific evolution is modeled by an encoder-decoder structure. We estimate the
evolution with an encoder-decoder neural network and give upper bounds for the
expected forecast error under specific structural and sparsity assumptions. The
results are shown separately for conditions either on the absolutely regular
mixing coefficients or the functional dependence measure of the observed
process. In a quantitative simulation we discuss the behavior of the network
estimator under different model assumptions. We corroborate our theory by a
real data example where we consider forecasting temperature data.
