Projection-based segregated, iterative, and multi-stage physics-informed neural network for incompressible Navier-stokes equations
Abstract
Physics-informed neural networks(PINNs) offer a mesh-free and continuously differentiable approach to fluid simulation. However, standard monolithic PINNs for incompressible Navier-Stokes (NS) equations often suffer from severe constraint coupling and spectral biases, limiting both convergence and accuracy. To address this challenge, we propose a segregated, iterative, and multi-stage PINN (SIM-PINN) inspired by the classical projection method in computational fluid dynamics. In SIM-PINN, the nonlinear NS equations are decoupled into the linearized momentum and pressure Poisson equations. Each is addressed separately using dedicated multi-stage neural networks and coupled through projection-based inner iterations. Each stage of the multi-stage neural networks for the momentum equations corresponds to one iteration of the projection method, in which the velocity fields are progressively refined at targeted frequencies. Upon iteration convergence, solutions to the fully coupled and nonlinear NS equations are recovered. By integrating the segregated and iterative framework with the multi-stage training, SIM-PINN alleviates the optimization difficulties of the incompressible NS equations arising from nonlinearity, constraint coupling, and spectral biases. The improved convergence and accuracy are demonstrated by simulations of steady lid-driven cavity and unsteady Kolmogorov flows.
Related
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BibTeX
@misc{zhang2026projectionbased,
title = {{Projection-based segregated, iterative, and multi-stage physics-informed neural network for incompressible Navier-stokes equations}},
author = {Zhenyu Zhang and Zhixiang Ling and Yinghao Huo and Huayong Liu and Yunde Su},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07387},
eprint = {2611.07387},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07387},
}
RIS
TY - UNPB
TI - Projection-based segregated, iterative, and multi-stage physics-informed neural network for incompressible Navier-stokes equations
AU - Zhenyu Zhang
AU - Zhixiang Ling
AU - Yinghao Huo
AU - Huayong Liu
AU - Yunde Su
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07387
UR - https://arxiv.org/abs/2611.07387
AB - Physics-informed neural networks(PINNs) offer a mesh-free and continuously differentiable approach to fluid simulation. However, standard monolithic PINNs for incompressible Navier-Stokes (NS) equations often suffer from severe constraint coupling and spectral biases, limiting both convergence and accuracy. To address this challenge, we propose a segregated, iterative, and multi-stage PINN (SIM-PINN) inspired by the classical projection method in computational fluid dynamics. In SIM-PINN, the nonlinear NS equations are decoupled into the linearized momentum and pressure Poisson equations. Each is addressed separately using dedicated multi-stage neural networks and coupled through projection-based inner iterations. Each stage of the multi-stage neural networks for the momentum equations corresponds to one iteration of the projection method, in which the velocity fields are progressively refined at targeted frequencies. Upon iteration convergence, solutions to the fully coupled and nonlinear NS equations are recovered. By integrating the segregated and iterative framework with the multi-stage training, SIM-PINN alleviates the optimization difficulties of the incompressible NS equations arising from nonlinearity, constraint coupling, and spectral biases. The improved convergence and accuracy are demonstrated by simulations of steady lid-driven cavity and unsteady Kolmogorov flows.
ER -