Whether listening to overlapping conversations in a crowded room or recording the simultaneous electrical activity of millions of neurons, the natural world abounds with sparse measurements of complex overlapping signals that arise from dynamical processes. While tools that separate mixed signals into linear sources have proven necessary and useful, the underlying equational forms of most natural signals are unknown and nonlinear. Hence, there is a need for a framework that is general enough to extract sources without knowledge of their generating equations and flexible enough to accommodate nonlinear, even chaotic, sources. Here, we provide such a framework, where the sources are chaotic trajectories from independently evolving dynamical systems. We consider the mixture signal as the sum of two chaotic trajectories and propose a supervised learning scheme that extracts the chaotic trajectories from their mixture. Specifically, we recruit a complex dynamical system as an intermediate processor that is constantly driven by the mixture. We then obtain the separated chaotic trajectories based on this intermediate system by training the proper output functions. To demonstrate the generalizability of this framework in silico, we employ a tank of water as the intermediate system and show its success in separating two-part mixtures of various chaotic trajectories. Finally, we relate the underlying mechanism of this method to the state-observer problem. This relation provides a quantitative theory that explains the performance of our method, and why separation is difficult when two source signals are trajectories from the same chaotic system.
