Solving and visualizing unsteady 3D Navier-Stokes equations with physics-informed neural networks

Abstract Physics-informed neural networks (PINNs) offer a promising mesh-free methodology for solving complex fluid dynamics problems governed by partial differential equations. This study develops and implements a PINN framework to solve the unsteady three-dimensional (3D) incompressible Navier–Stokes equations within a unit cubic domain. The network is trained using DeepXDE to accurately predict the velocity and pressure fields across both spatial and temporal domains, subject to divergence-free initial condition, and no-slip boundary conditions. The training process achieves high fidelity, evidenced by a composite loss converging to values on the order of 10 −8 , indicating strong adherence to the governing physics. Advanced visualization techniques utilizing PyVista enable the extraction and rendering of dense streamlines and vortical structures directly from the continuous PINN solution, capturing the transient evolution of complex flow features without reliance on conventional grid-based methods. Results demonstrate the framework’s capability to resolve intricate flow phenomena, including the formation and interaction of coherent structures, vortex stretching, and the temporal decay of kinetic energy due to viscous dissipation. Quantitative analyses of vorticity magnitude evolution, Q-criterion isosurfaces, probability density functions, energy spectra, and proper orthogonal decomposition further validate the solution’s physical consistency. This work establishes a robust PINN-based workflow for high-dimensional, time-dependent fluid simulations, providing a scalable alternative to traditional computational fluid dynamics that integrates solving with advanced mesh-free visualization for detailed exploration of complex 3D flow dynamics. A concise description of PINNs, which represent the solution fields by neural-network outputs while enforcing the governing partial differential equations, boundary and initial conditions directly during training has been provided