Title: Forecasting causal dynamics with universal reservoirs

Abstract: An iterated multistep forecasting scheme based on recurrent neural networks (RNN) is proposed for the time series generated by causal chains with infinite memory. This forecasting strategy contains, as a particular case, the iterative prediction strategies for dynamical systems that are customary in reservoir computing. Readily computable error bounds are obtained as a function of the forecasting horizon, functional and dynamical features of the specific RNN used, and the approximation error committed by it. The framework in the paper circumvents difficult-to-verify embedding hypotheses that appear in previous references in the literature and applies to new situations like the finite-dimensional observations of functional differential equations or the deterministic parts of stochastic processes to which standard embedding techniques do not necessarily apply.