This article introduces a neural network-based signal processing framework for intelligent Reflecting surface-aided (IRS) wireless communications systems. By modeling radio frequency (RF) impairments inside the meta-atoms of IRS - including nonlinearity and memory effects) - we present an approach that generalizes the entire IRS-aided system as a reservoir computing (RC) system, an efficient recurrent neural network (RNN) operating in a state near the “edge of chaos.” This framework enables us to take advantage of the nonlinearity of this “fabricated” wireless environment to overcome link degradation due to model mismatch. Accordingly, the randomness of the wireless channel and RF imperfections are naturally embedded into the RC framework, enabling the internal RC dynamics lying on the edge of chaos. Furthermore, several practical issues, such as channel state information acquisition, passive beamforming design, and physical layer reference signal design, are discussed.