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Physics-informed deep learning for incompressible laminar flows
Chengping Rao, Hao Sun, Yang Liu
TL;DR
Reliable fluid-flow simulation with deep learning is difficult when rich simulation data are costly, motivating physics-informed methods that embed governing laws. This paper develops a mixed-variable PINN using continuum and constitutive formulations plus a stream function, and reports accurate, trainable predictions for steady and transient low-Reynolds-number laminar flows. The approach agrees well with reference numerical solutions, while its demonstrated applications remain limited to low-Reynolds-number laminar cases.
Problem
Obtaining rich datasets for reliable fluid-flow models is computationally expensive, motivating physics-informed learning to reduce dependence on data.
Method
A mixed-variable PINN uses continuum and constitutive equations, a stream function, and physics residual losses to simulate incompressible laminar flows without measurement data.
Results
The mixed-variable scheme improves reported PINN accuracy and trainability, with steady and transient predictions agreeing well with reference numerical solutions.
Takeaways & Limitations
The proposed scheme demonstrates potential for accurate physics-informed simulation of steady and transient incompressible laminar flows at low Reynolds numbers.
Takeaways & Limitations
The applications are limited to laminar flows at low Reynolds numbers, while finer collocation for larger-Reynolds-number cases would increase memory use and computational cost.
Abstract
from arXiv · showhide
Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems, whose basic concept is to embed physical laws to constrain/inform neural networks, with the need of less data for training a reliable model. This can be achieved by incorporating the residual of physics equations into the loss function. Through minimizing the loss function, the network could approximate the solution. In this paper, we propose a mixed-variable scheme of physics-informed neural network (PINN) for fluid dynamics and apply it to simulate steady and transient laminar flows at low Reynolds numbers. A parametric study indicates that the mixed-variable scheme can improve the PINN trainability and the solution accuracy. The predicted velocity and pressure fields by the proposed PINN approach are also compared with the reference numerical solutions. Simulation results demonstrate great potential of the proposed PINN for fluid flow simulation with a high accuracy.
1. Introduction
Physics-informed deep learning embeds governing physics into neural-network training to reduce reliance on costly simulation data. This paper applies a mixed-variable PINN to steady and transient low-Reynolds-number laminar flows without measurement data.
- Motivation: Physics-informed deep learning embeds governing PDEs and initial/boundary conditions into neural networks through residual-based constraints.The approach aims to reduce dependence on dataset quality and quantity while improving robustness and interpretability.
- Motivation: The proposed study addresses the computational expense of obtaining rich simulation datasets for reliable deep-learning models.
- Related work: PINNs extend physics-informed learning by also accommodating measurement data when available, enabling modeling of partially understood systems.
- Contribution: The paper formulates a mixed-variable PINN for viscous incompressible laminar flows without any measurement data.It evaluates steady and transient flow past a circular cylinder and compares accuracy and trainability with a traditional scheme.
2. Methodology
The methodology reformulates incompressible Navier–Stokes dynamics into a mixed-variable PINN using continuum and constitutive equations. It uses a stream function and residual-based physics losses to enforce physical constraints and improve trainability.
- Mixed-variable formulation: The Navier–Stokes formulation is converted into continuum and constitutive equations because its original residual contains multiple latent variables and high-order derivatives.
- Mixed-variable formulation: The continuum-mechanics formulation reduces derivative order and improves DNN trainability in the reported numerical comparisons.
- Network outputs: The DNN maps spatiotemporal inputs to stream function, pressure, and stress, with the stream function ensuring divergence-free velocity.For two-dimensional problems, velocity is computed from the stream function through a curl operation, so continuity is satisfied automatically.
- Loss construction: The loss combines optional data loss with physics loss, while physics loss sums governing-equation and initial/boundary-condition losses.
3. Results
The proposed mixed-variable PINN is evaluated on steady and transient incompressible laminar flow past a circular cylinder. It reproduces reference fields and pressure histories while improving accuracy and trainability relative to the traditional scheme.
- Simulation setup: The study models steady and transient circular-cylinder flows with prescribed inlet, outlet, no-slip wall, and cylinder boundary conditions.Gravity is ignored; the steady case uses low-Reynolds-number conditions and collocation points refined near the cylinder.
- Steady flow: The 8 × 40 network achieves the best velocity-field result among tested configurations, and the mixed-variable PINN improves accuracy over the traditional PINN.The comparison uses relative ℓ2 velocity errors with β = 2.
- Steady flow: The mixed-variable scheme reproduces steady velocity and pressure fields, whereas the traditional scheme fails to enforce non-slip conditions on the lower and upper boundaries.Both PINNs use the same hyperparameters and collocation points in the field comparison, with ANSYS Fluent as reference.
- Steady flow: The mixed-variable PINN agrees well with ANSYS Fluent for pressure distribution on the cylinder surface.Cylinder-surface pressure is relevant to computing resultant drag and lift forces.
- Steady flow: Across β values, the mixed-variable scheme yields consistent convergence, while the traditional scheme is significantly affected by the boundary-condition-loss weight.The paper attributes the improvement to the reduced derivative order required to construct the loss, making optimization easier.
- Transient flow: The proposed PINN predicts transient flow evolution and pressure time histories that match ANSYS Fluent well.The transient case uses a time-varying parabolic inlet and pressure probes P1, P2, and P3 on the cylinder surface.
4. Conclusions
The paper proposes a mixed-variable PINN for incompressible laminar flows, combining continuum and constitutive equations with a stream function. It reports high accuracy and good trainability, while applications remain limited to low-Reynolds-number laminar flows.
- The proposed scheme models incompressible laminar flows using continuum equations, constitutive laws, and a stream function in a mixed-variable PINN.
- The comparison study indicates that the mixed-variable scheme provides high accuracy and good trainability.
- The PINN results agree well with reference numerical solutions for both steady and transient flow cases.
- Applications are limited to low-Reynolds-number laminar flows, although the approach is theoretically applicable to turbulent flows at larger Reynolds numbers.Finer collocation discretization for such cases could cause memory issues and substantially increase computational cost.