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

TL;DR

The paper addresses solving 2D incompressible Navier–Stokes equations using hp-VPINNs in forward and inverse settings on complex geometries. It extends FastVPINNs to vector-valued problems by formulating variational residuals and employing tensor-based, GPU-accelerated loss computations, achieving a reported 2× reduction in training time relative to PET PINN baselines in studied cases. The authors validate the approach on multiple benchmarks, including Kovasznay flow, lid-driven cavity, flow through a channel, backward-facing step, Falkner–Skan boundary layer, and flow past a cylinder, and demonstrate successful Reynolds-number identification from sensor data. This combination of efficiency, accuracy, and geometric flexibility broadens the applicability of hp-VPINNs for practical CFD tasks and parameter estimation problems.

Abstract

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.

Figures (20)

• Figure 1: FastVPINNs Tensor schematic representation for residual computation.
• Figure 2: Solution of Burgers' equation: Solution predicted by FastVPINNs, exact solution and point-wise errors for $u$ and $v$ respectively.
• Figure 3: Training for Kovasznay flow: (a) PDE loss, Dirichlet boundary loss and total loss during training (b) PDE loss split into its components: x-momentum residual, y-momentum residual and divergence loss.
• Figure 4: Solution of Kovasznay flow for $\text{Re}=40$: Solution predicted by FastVPINNs, exact solution and point-wise errors for $u$, $v$ and $p$ respectively.
• Figure 5: Kovasznay Flow: Relative errors of $u$, $v$ and $p$ for various grid sizes