The variational approach of finding the extremum of an objective functional is an alternative approach to the solution of partial differential equations in mechanics of continua. The great challenge in the calculus of variations direct methods is to find a set of functions that will be able to approximate the solution accurately enough. Artificial neural networks are a powerful tool for approximation, and the physics-based functional can be the natural loss for a machine learning method. In this paper, we focus on the loss that may take non-linear fluid properties and mass forces into account. We modified the energy-based variational principle and determined the constraints on its unknown functions that implement boundary conditions. We explored artificial neural networks as an option for loss minimization and the approximation of the unknown functions. We compared the obtained results with the known solutions. The proposed method allows modeling non-Newtonian fluids flow including blood, synthetic oils, paints, plastic, bulk materials, and even rheomagnetic fluids. The fluids flow velocity approximation error was up to 4% in comparison with the analytical and numerical solutions.