Unsupervised simulation of incompressible flows with physics- and equality- constrained artificial neural networks

Abstract

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations, yet their success in simulating incompressible flows at high Reynolds numbers remains limited. Existing approaches rely on auxiliary labeled data, supervised pretraining, or reference solutions, and no purely unsupervised method comparable to conventional finite-difference or finite-volume solvers has been demonstrated. We attribute this gap to the absence of a mechanism for enforcing the divergence-free constraint and boundary conditions to strict tolerances. To address this, we adopt the physics- and equality-constrained artificial neural network (PECANN) framework with a conditionally adaptive augmented Lagrangian method (CA-ALM), and introduce a pressure-Poisson-based objective. The residual of the pressure Poisson equation is minimized subject to the momentum and continuity equations and boundary conditions on the primitive variables as equality constraints, with CA-ALM enforcing all constraints tightly. For advection-dominated, high-Reynolds-number flows, we further propose an adaptive vanishing entropy viscosity that stabilizes early training without influencing the converged solution. A baseline that instead uses the momentum residual as the objective proves ineffective under the same machinery, underscoring the critical role of the pressure-Poisson objective. The method is assessed on lid-driven cavity flow up to $Re=7{,}500$, three-dimensional unsteady Beltrami flow, and steady and unsteady flow past a circular cylinder with general inflow-outflow boundary conditions, including an ablation study identifying admissible outlet conditions -- all without labeled data or supervised pretraining. Notably, it captures the spontaneous onset of periodic vortex shedding in unsteady cylinder flow without external perturbations, starting from a randomly initialized network.