Variational Physics-Informed Neural Networks for Unsteady Incompressible Flows

Recent advances in Physics-Informed Neural Networks (PINNs) applied to fluid mechanics have largely relied on the Newtonian framework, utilizing Navier–Stokes equations or their derivatives to train the neural network. Here, we propose an alternative approach based on variational methods; specifically employing the principle of minimum pressure gradient to turn the fluid mechanics problem into a minimization problem whose solution can be used to predict the flow field in unsteady incompressible viscous flows.

We apply this novel approach to study the unsteady flow field in a lid-driven cavity at Reynolds numbers ranging from 100 to 5000. The computational performance of the proposed method is compared to conventional PINNs showcasing its accuracy and efficiency. Computational results indicate that the variational approach to PINNs offers a robust and efficient way to solve incompressible fluid mechanics problems. Moreover, it exhibits the potential for extension to turbulent and non-Newtonian fluids, paving the way for broader implementations in fluid mechanics.