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

TL;DR

This work introduces a meshless, physics- and equality-constrained neural solver (PECANN) for incompressible flows, using a single network to predict velocity and pressure while constraining the pressure Poisson equation through momentum and continuity along with velocity-boundary conditions. The training relies on a CA-ALM framework with adaptive entropy viscosity and a Fourier feature mapping to stabilize and enhance expressivity, eliminating the need for labeled data. Demonstrations on 2D lid-driven cavity flows up to $Re=7500$ and steady cylinder flow at $Re=40$ show high fidelity to benchmark solutions, with superior performance of the pressure-based formulation over a baseline momentum-residual approach. The approach offers a principled, meshless alternative for incompressible flow simulation and has potential for time-dependent problems and inverse analyses.

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.

Figures (13)

• Figure 1: Lid-driven cavity flow at $Re = 2500$: mean and standard deviation (Std. band) of the predicted $v$-velocity along $y = 0.5$ for (a) the baseline formulation (Eq. \ref{['eq:baseline_constrained_flow_problem']}) and (b) the proposed pressure-based formulation (Eq. \ref{['eq:proposed_constrained_flow_problem']}).
• Figure 2: Lid-driven cavity flow at $Re = 2500$: mean and standard deviation (Std. band) of the predicted $u$-velocity along $x = 0.5$ for (a) the baseline formulation (Eq. \ref{['eq:baseline_constrained_flow_problem']}) and (b) the proposed pressure-based formulation (Eq. \ref{['eq:proposed_constrained_flow_problem']}).
• Figure 3: Lid-driven cavity flow at $Re = 2500$: (a) predicted velocity field with streamlines from one trial using adaptive, vanishing entropy viscosity, $\nu_a^a$, in the pressure-based formulation, and (b) the corresponding pressure field.
• Figure 4: Lid-driven cavity flow at $Re = 2500$: comparison of the training behaviors between the proposed and baseline formulations using the Fourier net with an adaptive, vanishing entropy viscosity, $\nu_a^a$. (a) Evolution of $\nu_a^a$ over epochs for all trials, (b) evolution of the objective and constraint losses from one trial of the proposed formulation, and (c) evolution of the loss terms in the baseline formulation.
• Figure 5: Lid-driven cavity flow at $Re = 2500$: (a) comparison of linear entropy viscosity $\nu_a^l$ evolution between the Fourier net and MLP under the pressure-based formulation, (b) evolution of the objective and constraint losses from a Fourier-net trial, and (c) the corresponding predicted velocity field.