Virtual viscometer based on Physics-Informed mCRE formulation using Deep-Learning with Finite-Volume Multiple-Mesh Training. Part I: steady flow CNN-augmented Reduced-Order Model

Fluid rheology law identification requires viscometers in order to identify material parameters characterizing kinematic properties. Among these parameters the viscosity is a key factor since it is associated with shear stress effects and describes resistance of liquids to fluid flow according to friction occuring inside the medium during motion. However identification of viscosity often remains a difficulty due to the noise disturbing measurements that have to be filtered in a way to be defined. Moreover parameter inverse identification is most of the time associated with ill-posed problems resulting in a non-unicity of solutions, especially in the presence of strong shear effects. In the present work, a Physics-Informed Neural Network Architecture (mCRE-based PINN) is introduced as a Reduced-Order Model (ROM) in order to create a virtual viscometer in order to deal with viscosity identification in homogeneous and heterogeneousenvironments. A modified Constitutive-Relation Error (mCRE) is involved in order to enforce structural physical consistency of solutions and the proposed approach relies on a Convolutional Neural Network (CNN) that projects noisy velocity fields onto a low-dimensional space of physically admissible solutions. To ensure robust parameter identification, the modified mCRE framework acts as a physical constraint within the Artificial Neural Network (NN) Architecture. Unlike purely Data-Driven Methods (DDM), this hybrid approach provides a Physics-Informed ROM of fluid behavior that can be used as a virtual viscometer by minimizing the constitutive residual, effectively filtering out data noise while enforcing the underlying physical laws. A parameter sensitivity analysis is performed here in order to highlight the strength of the mCRE-based neural operator providing better accuracy and robustness for inverse viscosity identification than standard regression-based ROMs. The way Deep Learning (DL) is combined with physical constraints provides a powerful approach towards more reliable and interpretable surrogate modelling that may be appropriate in the context of Computational Fluid Dynamics (CFD). This new generation of Physics-Informed NN-based ROM will make it possible in a near future to gain efficiency for in-situ real-time flow field monitoring and control thanks to robust optimized multi-parameter meta-models.