Artificial neural networks as a natural tool in solution of variational problems in hydrodynamics

Artificial neural networks are a powerful tool for spatial and temporal functions approximation. This study introduces a novel approach for modeling non-Newtonian fluid flows by minimizing a proposed power loss metric, which aligns with the variational formulation of boundary value problems in hydrodynamics and extends the classical Lagrange variational principle. The method is distinguished by its data-free nature, enabling problem-solving through 2D or 3D images of the flow domain. Validation was performed using both multi-layer perceptrons and U-Net architectures, with results compared against analytical and numerical benchmarks. The method demonstrated good results with a relative error of 1.41% in comparison with the analytical solution for non-Newtonian fluids. The power loss formulation offers a clear advantage by simplifying the modeling process and enhancing interpretability. Notably, the proposed method demonstrates improvements over existing techniques by providing algorithmic simplicity and universality, with applications ranging from blood flow modeling in vessels and tissues to broader hydrodynamic scenarios.