Bridging Computational Fluid Dynamics Algorithm and Physics-Informed Learning: SIMPLE-PINN for Incompressible Navier-Stokes Equations

2026-03-25

AI Summary

This work proposes SIMPLE-PINN, a novel physics-informed neural network framework that addresses the challenges of training instability and slow convergence commonly encountered by traditional PINNs when solving high-Reynolds-number nonlinear fluid flow problems. By integrating the pressure–velocity coupling concept from the SIMPLE algorithm in computational fluid dynamics into the PINN architecture, the method introduces a velocity–pressure correction loss term to strengthen the physical consistency and inter-variable coordination imposed by the Navier–Stokes equations. A hybrid numerical–automatic differentiation strategy is further employed to enhance generalization in complex geometries. Requiring no data-driven supervision, SIMPLE-PINN achieves high-accuracy solutions in just 448 seconds for the lid-driven cavity flow at Re = 20,000 and accurately captures the full vortex shedding dynamics over t = 0–100 in flow past a cylinder, markedly improving both convergence speed and predictive accuracy.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs) by directly embedding them into the loss function. Despite their notable success, existing PINNs often exhibit training instability and slow convergence when applied to strongly nonlinear fluid dynamics problems. To address these challenges, this paper proposes a novel PINN framework, named as SIMPLE-PINN, which incorporates velocity and pressure correction loss terms inspired by the semi-implicit pressure link equation. These correction terms, derived from the momentum and continuity residuals, are tailored for the PINN framework, ensuring velocity-pressure coupling and reinforcing the underlying physical constraints of the Navier-Stokes equations. Through this, the framework can effectively mitigate training instability and accelerate convergence to achieve accurate solution. Furthermore, a hybrid numerical-automatic differentiation strategy is employed to improve the model's generalizability in resolving flows involving complex geometries. The performance of SIMPLE-PINN is evaluated on a range of challenging benchmark cases, including strongly nonlinear flows, long-term flow prediction, and multiphysics coupling problems. The numerical results demonstrate SIMPLE-PINN's high accuracy and rapid convergence. Notably, SIMPLE-PINN achieves, for the first time, a fully data-free solution of lid-driven cavity flow at Re=20000 in just 448s, and successfully captures the onset and long-time evolution of vortex shedding in flow past a cylinder over t=0-100. These findings demonstrate SIMPLE-PINN's potential as a reliable and competitive neural solver for complex PDEs in intelligent scientific computing, with promising engineering applications in aerospace, civil engineering, and mechanical engineering.

Problem

• Physics-informed neural networks
• Training instability
• Slow convergence
• Incompressible Navier-Stokes equations
• Fluid dynamics

Innovation

• SIMPLE-PINN
• velocity-pressure coupling
• physics-informed neural networks
• Navier-Stokes equations
• hybrid differentiation strategy