Several studies have explored the idea of deep neural networks as surrogates for nonlinear model predictive control. However, the effect of the approximation on the structure and the behavior of a tube model predictive controller has not been thoroughly analyzed. This work investigates this effect using ideas from statistical machine learning and contraction theory, with particular emphasis on systems whose dynamics are contractive. The approximation error is considered as a perturbation on the control and is bounded probabilistically in terms of the network's generalisation error. The study is conducted within the ERC Consolidator Grant project DREAM-ON with a target application to Structural Health Monitoring. The proposed methodology contributes to the feedback loop created between a physical asset and its digital twin. Numerical validation is provided using a simple two-dimensional nonlinear oscillator.