  The nudging data assimilation algorithm is a powerful tool used to forecast
phenomena of interest given incomplete and noisy observations. Machine learning
is becoming increasingly popular in data assimilation given its ease of
computation and forecasting ability. This work proposes a new approach to data
assimilation via machine learning where Deep Neural Networks (DNNs) are being
taught the nudging algorithm. The accuracy of the proposed DNN based algorithm
is comparable to the nudging algorithm and it is confirmed by the Lorenz 63 and
Lorenz 96 numerical examples. The key advantage of the proposed approach is the
fact that, once trained, DNNs are cheap to evaluate in comparison to nudging
where typically differential equations are needed to be solved. Standard
exponential type approximation results are established for the Lorenz 63 model
for both the continuous and discrete in time models. These results can be
directly coupled with estimates for DNNs (whenever available), to derive the
overall approximation error estimates of the proposed algorithm.
