This paper carries out sparse-penalized deep neural networks predictors for learning weakly dependent processes, with a broad class of loss functions. We deal with a general framework that includes, regression estimation, classification, times series prediction, $\cdots$ The $ψ$-weak dependence structure is considered, and for the specific case of bounded observations, $θ_\infty$-coefficients are also used. In this case of $θ_\infty$-weakly dependent, a non asymptotic generalization bound within the class of deep neural networks predictors is provided. For learning both $ψ$ and $θ_\infty$-weakly dependent processes, oracle inequalities for the excess risk of the sparse-penalized deep neural networks estimators are established. When the target function is sufficiently smooth, the convergence rate of these excess risk is close to $\mathcal{O}(n^{-1/3})$. Some simulation results are provided, and application to the forecast of the particulate matter in the Vitória metropolitan area is also considered.