Physically plausible and conservative solutions to Navier–Stokes equations using physics-informed CNNs

The physics-informed neural network (PINN) is an emerging approach for efficiently solving partial differential equations (PDEs) using neural networks. The physics-informed convolutional neural network (PICNN), a variant of PINN enhanced by convolutional neural networks (CNNs), has achieved better results on a series of PDEs since the parameter-sharing property of CNNs is effective in learning spatial dependencies. However, applying existing PICNN-based methods to solve Navier–Stokes equations can generate oscillating predictions, which are inconsistent with the laws of physics and the conservation properties. To address this issue, we propose a novel method that combines PICNN with the finite volume method to obtain physically plausible and conservative solutions to Navier–Stokes equations. We derive the second-order upwind difference scheme of Navier–Stokes equations using the finite volume method. Then we use the derived scheme to calculate the partial derivatives and construct the physics-informed loss function. The proposed method is assessed by experiments on steady-state Navier–Stokes equations under different scenarios, including convective heat transfer and lid-driven cavity flow. The experimental results demonstrate that our method can effectively improve the plausibility and accuracy of the predicted solutions from PICNN.