Exploring the potential of physics-informed neural networks (PINN) in two-dimensional Couette-Poiseuille flow of newtonian and Bingham fluids

Abstract

This study aimed to develop, validate, and analyze the performance of machine learning (ML) techniques based on artificial neural networks (ANNs), with a focus on Physics-Informed Neural Networks (PINNs): An approach to approximate partial differential equations (PDE) solutions by training a neural ansatz to minimize the residuals of the governing equations and boundary/initial conditions, without requiring labeled data. PINNs stand out by eliminating the need for mesh discretization, experimental datasets, domain-specific software, and even allowing for extrapolation beyond the training region, positioning themselves as a promising alternative to traditional numerical methods.In this work, the Couette–Poiseuille flow of Newtonian and Bingham fluids was investigated, a configuration relevant to various industrial contexts, particularly in drilling engineering and fluid transport. The study was structured around two PINN model stages: an initial exploratory base model and a final refined model. For Newtonian fluids, the final model demonstrated high accuracy in reconstructing the velocity and pressure fields, rigorously respecting the imposed boundary conditions and showing excellent agreement with reference solutions obtained by the Finite Volume Method (FVM) and, when available, exact solution. In contrast, for Bingham fluids, the results revealed the limitations of the approach in dealing with stronger nonlinearities and internal discontinuities typical of yield-stress fluids, even when applying Papanastasiou’s regularization. In such cases, a significant increase in error and loss of physical coherence was observed, including partial collapse or inversion of the predicted velocity and pressure fields. Despite these difficulties, the PINNs proved structurally robust, maintaining key features such as flow symmetry, general flow trends, and partial compliance with boundary conditions even under critical regimes. This work adopted the classical PINN formulation, without incorporating adaptive loss balancing, domain decomposition, hard constraints, or other advanced techniques, limitations imposed by the capabilities of the library used, DeepXDE. Nevertheless, the results provided valuable insights into the main challenges associated with applying PINNs to problems with simple geometries but complex rheological behavior.