Structure-preserving neural networks in data-driven rheological models

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.