  New algorithms called nudging induced neural networks (NINNs), to control and
improve the accuracy of deep neural networks (DNNs), are introduced. The NINNs
framework can be applied to almost all pre-existing DNNs, with forward
propagation, with costs comparable to existing DNNs. NINNs work by adding a
feedback control term to the forward propagation of the network. The feedback
term nudges the neural network towards a desired quantity of interest. NINNs
offer multiple advantages, for instance, they lead to higher accuracy when
compared with existing data assimilation algorithms such as nudging. Rigorous
convergence analysis is established for NINNs. The algorithmic and theoretical
findings are illustrated on examples from data assimilation and chemically
reacting flows.
