Solution of Incompressible Flow Equations with Physics and Equality Constrained Artificial Neural Networks

Abstract: We present a meshless method for the solution of incompressible Navier-Stokes equations in advection-dominated regimes using physics- and equality-constrained artificial neural networks combined with a conditionally adaptive augmented Lagrangian formulation. A single neural network parameterizes both the velocity and pressure fields, and is trained by minimizing the residual of a Poisson's equation for pressure, constrained by the momentum and continuity equations, together with boundary conditions on the velocity field. No boundary conditions are imposed on the pressure field aside from anchoring the pressure at a point to prevent its unbounded development. The training is performed from scratch without labeled data, relying solely on the governing equations and constraints. To enhance accuracy in advection-dominated flows, we employ a single Fourier feature mapping of the input coordinates. The proposed method is demonstrated for the canonical lid-driven cavity flow up to a Reynolds number of 7,500 and for laminar flow over a circular cylinder with inflow-outflow boundary conditions, achieving excellent agreement with benchmark solutions. We further compare the present formulation against alternative objective-function constructions based on different arrangements of the flow equations, thereby highlighting the algorithmic advantages of the proposed formulation centered around the Poisson's equation for pressure.