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.

翻译：本文探讨了在斯托克斯流中，采用保结构神经网络替代通常用于描述非牛顿流体流变性的解析物理模型的重要性和影响。具体而言，我们提出并基于实际场景测试了一种构建数据驱动流变模型的新策略，该策略基于凸输入输出神经网络（ICNNs）——一类标量值前馈神经网络，其输出相对于输入具有凸性。此外，通过一系列详细的数值实验，我们证明了使用ICNNs对于确保相关非牛顿斯托克斯微分问题的适定性至关重要。最后，基于非牛顿斯托克斯问题的一个新扰动结果，我们研究了基于数据驱动ICNN的流变模型对有限元近似精度的影响。