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

arXiv: https://arxiv.org/abs/2511.18820 · HTML: https://mlsanity.com/paper/2511.18820/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 55 references that are in the map.
- First posted on arXiv 24 Nov 2025.

## Builds on (most cited first, 7 of 7)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks](https://mlsanity.com/paper/2409.13644/) (2024)

## Abstract (arXiv)

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.