  Recent developments on deep learning established some theoretical properties
of deep neural networks estimators. However, most of the existing works on this
topic are restricted to bounded loss functions or (sub)-Gaussian or bounded
input. This paper considers robust deep learning from weakly dependent
observations, with unbounded loss function and unbounded input/output. It is
only assumed that the output variable has a finite $r$ order moment, with $r
>1$. Non asymptotic bounds for the expected excess risk of the deep neural
network estimator are established under strong mixing, and $\psi$-weak
dependence assumptions on the observations. We derive a relationship between
these bounds and $r$, and when the data have moments of any order (that is
$r=\infty$), the convergence rate is close to some well-known results. When the
target predictor belongs to the class of H\"older smooth functions with
sufficiently large smoothness index, the rate of the expected excess risk for
exponentially strongly mixing data is close to or as same as those for obtained
with i.i.d. samples. Application to robust nonparametric regression and robust
nonparametric autoregression are considered. The simulation study for models
with heavy-tailed errors shows that, robust estimators with absolute loss and
Huber loss function outperform the least squares method.
