An efficient hp-Variational PINNs framework for incompressible Navier-Stokes equations

Physics-informed neural networks (PINNs) are able to solve partial differential equations (PDEs) by incorporating the residuals of the PDEs into their loss functions. Variational Physics-Informed Neural Networks (VPINNs) and hp-VPINNs use the variational form of the PDE residuals in their loss function. Although hp-VPINNs have shown promise over traditional PINNs, they suffer from higher training times and lack a framework capable of handling complex geometries, which limits their application to more complex PDEs. As such, hp-VPINNs have not been applied in solving the Navier-Stokes equations, amongst other problems in CFD, thus far. FastVPINNs was introduced to address these challenges by incorporating tensor-based loss computations, significantly improving the training efficiency. Moreover, by using the bilinear transformation, the FastVPINNs framework was able to solve PDEs on complex geometries. In the present work, we extend the FastVPINNs framework to vector-valued problems, with a particular focus on solving the incompressible Navier-Stokes equations for two-dimensional forward and inverse problems, including problems such as the lid-driven cavity flow, the Kovasznay flow, and flow past a backward-facing step for Reynolds numbers up to 200. Our results demonstrate a 2x improvement in training time while maintaining the same order of accuracy compared to PINNs algorithms documented in the literature. We further showcase the framework's efficiency in solving inverse problems for the incompressible Navier-Stokes equations by accurately identifying the Reynolds number of the underlying flow. Additionally, the framework's ability to handle complex geometries highlights its potential for broader applications in computational fluid dynamics. This implementation opens new avenues for research on hp-VPINNs, potentially extending their applicability to more complex problems.