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This paper carries out sparse-penalized deep neural networks predictors forlearning weakly dependent processes, with a broad class of loss functions. Wedeal with a general framework that includes, regression estimation,classification, times series prediction, $\cdots$ The $\psi$-weak dependencestructure is considered, and for the specific case of bounded observations,$\theta_\infty$-coefficients are also used. In this case of$\theta_\infty$-weakly dependent, a non asymptotic generalization bound withinthe class of deep neural networks predictors is provided. For learning both$\psi$ and $\theta_\infty$-weakly dependent processes, oracle inequalities forthe excess risk of the sparse-penalized deep neural networks estimators areestablished. When the target function is sufficiently smooth, the convergencerate of these excess risk is close to $\mathcal{O}(n^{-1/3})$. Some simulationresults are provided, and application to the forecast of the particulate matterin the Vit\'{o}ria metropolitan area is also considered.

Penalized deep neural networks estimator with general loss functions under weak dependence