  We devise a machine learning technique to solve the general problem of
inferring network links that have time-delays. The goal is to do this purely
from time-series data of the network nodal states. This task has applications
in fields ranging from applied physics and engineering to neuroscience and
biology. To achieve this, we first train a type of machine learning system
known as reservoir computing to mimic the dynamics of the unknown network. We
formulate and test a technique that uses the trained parameters of the
reservoir system output layer to deduce an estimate of the unknown network
structure. Our technique, by its nature, is non-invasive, but is motivated by
the widely-used invasive network inference method whereby the responses to
active perturbations applied to the network are observed and employed to infer
network links (e.g., knocking down genes to infer gene regulatory networks). We
test this technique on experimental and simulated data from delay-coupled
opto-electronic oscillator networks. We show that the technique often yields
very good results particularly if the system does not exhibit synchrony. We
also find that the presence of dynamical noise can strikingly enhance the
accuracy and ability of our technique, especially in networks that exhibit
synchrony.
