  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 \emph{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, such as why separation is difficult
when two source signals are trajectories from the same chaotic system.
