  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 trade-off 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.
