  In this paper we address the importance and the impact of employing structure
preserving neural networks as surrogate of the analytical physics-based models
typically employed to describe the rheology of non-Newtonian fluids in Stokes
flows. In particular, we propose and test on real-world scenarios a novel
strategy to build data-driven rheological models based on the use of
Input-Output Convex Neural Networks (ICNNs), a special class of feedforward
neural network scalar valued functions that are convex with respect to their
inputs. Moreover, we show, through a detailed campaign of numerical
experiments, that the use of ICNNs is of paramount importance to guarantee the
well-posedness of the associated non-Newtonian Stokes differential problem.
Finally, building upon a novel perturbation result for non-Newtonian Stokes
problems, we study the impact of our data-driven ICNN based rheological model
on the accuracy of the finite element approximation.
