Three-dimensional laminar flow using physics informed deep neural networks

Physics informed neural networks (PINNs) have demonstrated their effectiveness in solving partial differential equations (PDEs). By incorporating the governing equations and boundary conditions directly into the neural network architecture with the help of automatic differentiation, PINNs can approximate the solution of a system of PDEs with good accuracy. Here, an application of PINNs in solving three-dimensional (3D) Navier–Stokes equations for laminar, steady, and incompressible flow is presented. Notably, our approach involves deploying PINNs using feed-forward deep neural networks (DNNs) without depending on any simulation or experimental data. This investigation focuses on 3D square channel flow and 3D lid-driven cavity flow. For each case, one deep neural network was trained using only the governing equations and boundary conditions. Finally, the PINNs' results were compared with the computational fluid dynamics results. The goal was to assess the ability of PINNs (with DNN architectures) to predict the solution of Navier–Stokes equations in the 3D domain without any simulation or experimental data (unsupervised learning).