Signal demixing using multi-delay multi-layer reservoir computing

Brain circuitry involves a large number of recurrent feedback loops whose dynamics depend on interaction delays. Brain-inspired reservoir computing leverages the rich recurrent dynamics of interconnected units for performing tasks on inputs. In particular, time-delay reservoir computing uses the high-dimensional transient dynamics in nonlinear delayed feedback loop architectures for e.g. time series prediction and speech classification. The modification of the dynamical properties of delay-differential systems through the inclusion of multiple delays has also recently been shown to improve the performance of time-delay reservoir computing. Here we explore another aspect of such neuro-inspired computing of fundamental and technological importance: the ability to separate and predict two signals in a mixture, where each has some intrinsic predictability due to its underlying dynamics. This is illustrated using multi-delay and multi-layer reservoir computing with chaotic input signal mixtures. In contrast to Independent Component Analysis and related unsupervised learning techniques, the context here consists in the parallel supervised learning of the dynamics for each signal in order to predict each of them beyond the training set. Further, the superposition of the chaotic signals into a single input channel adds to the difficulty of the task. We quantify and explain this performance with various signals emanating from both deterministic and stochastic systems. Additionally, we explore the architecture of deep time-delay reservoir computers. Our findings demonstrate that multi-delay reservoir computing can learn and predict the future of two superimposed deterministic signals. Prediction—and thus separation—accuracy can be significantly higher in single and multi-layer time-delay reservoir computing when the first layer contains multiple delays. Bandpass filtering of the mixed signal to remove lower and higher frequencies improved the prediction by a few percent. In some cases, paradoxically, increasing the proportion of one chaotic signal in the mixture can actually help the learning of another chaotic signal, and thus slightly improve its prediction.