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.

中文翻译：

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通过水箱进行有序混沌源分离。

无论是在拥挤的房间里聆听重叠的对话，还是记录数百万个神经元的同时电活动，自然世界中都充斥着对由动态过程产生的复杂重叠信号的稀疏测量。尽管已证明将混合信号分离为线性信号源的工具是必要且有用的，但大多数自然信号的基本方程式仍是未知且非线性的。因此，需要一种框架，该框架足够通用以在不了解源生成方程的情况下提取源并且足够灵活以容纳非线性甚至混乱的源。在这里，我们提供了这样一个框架，其中源是来自独立演化的动力学系统的混沌轨迹。我们将混合信号视为两个混沌轨迹的总和，并提出了一种监督学习方案，该方案从它们的混合中提取出混沌轨迹。具体来说，我们招募了一个复杂的动力系统作为中间处理器，该系统经常由混合物驱动。然后，通过训练适当的输出函数，基于此中间系统获得分离的混沌轨迹。为了演示该框架在计算机上的通用性，我们使用一个水箱作为中间系统，并展示了其在分离各种混沌轨迹的两部分混合物方面的成功。最后，我们将这种方法的基本机制与状态观察者问题联系起来。这种关系提供了一种定量理论，可以解释我们方法的性能，