A large variety of dynamical systems, such as chemical and biomolecular systems, can be seen as networks of nonlinear entities. Prediction, control, and identification of such nonlinear networks require knowledge of the state of the system. However, network states are usually unknown, and only a fraction of the state variables are directly measurable. The observability problem concerns reconstructing the network state from this limited information. Here, we propose a general optimization-based approach for observing the states of nonlinear networks and for optimally selecting the observed variables. Our results reveal several fundamental limitations in network observability, such as the tradeoff between the fraction of observed variables and the observation length on one side, and the estimation error on the other side. We also show that, owing to the crucial role played by the dynamics, purely graph-theoretic observability approaches cannot provide conclusions about one's practical ability to estimate the states. We demonstrate the effectiveness of our methods by finding the key components in biological and combustion reaction networks from which we determine the full system state. Our results can lead to the design of novel sensing principles that can greatly advance prediction and control of the dynamics of such networks.

中文翻译：

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非线性网络的状态观测和传感器选择

各种各样的动力系统，例如化学和生物分子系统，可以被视为非线性实体的网络。此类非线性网络的预测、控制和识别需要了解系统的状态。然而，网络状态通常是未知的，并且只有一小部分状态变量是可以直接测量的。可观测性问题涉及从有限的信息重建网络状态。在这里，我们提出了一种基于优化的通用方法，用于观察非线性网络的状态并最佳地选择观察到的变量。我们的结果揭示了网络可观测性的几个基本限制，例如观测变量的比例和一侧的观测长度以及另一侧的估计误差之间的权衡。我们还表明，由于动力学发挥的关键作用，纯图论可观测性方法无法提供有关估计状态的实际能力的结论。我们通过寻找生物和燃烧反应网络中的关键组件来确定整个系统状态，从而证明了我们方法的有效性。我们的研究结果可以引导新型传感原理的设计，从而极大地推进对此类网络动态的预测和控制。