Solving Incompressible Navier-Stokes Equations with Physics and Equality Constrained Artificial Neural Networks

Robust numerical methods such as finite-volume and spectral element approaches have long been established for solving the incompressible Navier–Stokes equations with high accuracy. In contrast, while physics-informed neural networks and their variants have shown promise for solving complex partial differential equations, their application to advection-dominated incompressible flows—particularly in general settings without auxiliary labeled data—has yielded limited success. To address this challenge, we propose a pressure-based algorithm grounded in the Physics and Equality Constrained Artificial Neural Networks (PECANN) framework. Our approach leverages the augmented Lagrangian method with a novel penalty parameter update strategy and incorporates a single Fourier feature mapping to enhance convergence and predictive accuracy. Benchmark evaluations demonstrate the effectiveness of the proposed method in learning steady-state solutions for lid-driven cavity flows at Reynolds numbers up to 2500 and flow over a cylinder at Re = 40. These results highlight the potential of PECANN framework for accurate, mesh-free modeling of incompressible flows in complex settings.