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NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
Xiaowei Jin, Shengze Cai, Hui Li, George Em Karniadakis
TL;DR
The paper asks whether PINNs can directly simulate turbulence and whether alternative Navier–Stokes formulations improve accuracy or training. It develops NSFnets using velocity-pressure and vorticity-velocity formulations, evaluates them on laminar and turbulent flows, and studies manual and dynamic loss weighting. The results report accurate simulations, better laminar accuracy for VV-NSFnet than VP-NSFnet, and improved performance for both with dynamic weights.
Problem
The paper investigates whether PINNs can simulate turbulence directly and whether the vorticity-velocity formulation can improve accuracy or training efficiency.
Method
NSFnets use PINNs with velocity-pressure or vorticity-velocity Navier–Stokes equations embedded in the loss, using automatic differentiation and no pressure data for VP-NSFnet.
Results
Dynamic loss weights improve the performance of both NSFnets, while VV-NSFnet achieves better accuracy than VP-NSFnet for laminar cases.
Takeaways & Limitations
NSFnets can simulate complex incompressible flows using either formulation, with accuracy improved by proper manual or dynamic loss-weight tuning.
Takeaways & Limitations
The efficiency of PINNs has to be improved significantly, and careful tuning of anisotropic weights may be needed for balanced accuracy.
Abstract
from arXiv · showhide
We employ physics-informed neural networks (PINNs) to simulate the incompressible flows ranging from laminar to turbulent flows. We perform PINN simulations by considering two different formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. We refer to these specific PINNs for the Navier-Stokes flow nets as NSFnets. Analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. These two different forms of the Navier-Stokes equations together with the initial and boundary conditions are embedded into the loss function of the PINNs. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without splitting the equations. We obtain good accuracy of the NSFnet simulation results upon convergence of the loss function, verifying that NSFnets can effectively simulate complex incompressible flows using either the VP or the VV formulations. We also perform a systematic study on the weights used in the loss function for the data/physics components and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. Our results suggest that the accuracy of NSFnets, for both laminar and turbulent flows, can be improved with proper tuning of weights (manual or dynamic) in the loss function.
1. Introduction
The paper develops NSFnets to simulate incompressible flows with velocity-pressure or vorticity-velocity Navier–Stokes formulations. It systematically evaluates these formulations and loss-function weighting, including direct PINN simulation of turbulent channel flow.
- Prior work: Existing neural-network approaches in fluid mechanics include turbulence closures, reduced-order models, flow prediction, and experimental-data analysis.These approaches span RANS and LES modeling, cylinder-wake prediction, POD-coefficient prediction, and PIV analysis.
- Research questions: The paper asks whether PINNs can simulate turbulence directly and whether vorticity-velocity formulations can improve accuracy or training efficiency.These questions extend prior PINN studies of comparatively low-Reynolds-number laminar flows in velocity-pressure form.
- Evaluation: The study evaluates analytical two- and three-dimensional flows and DNS-based turbulent channel flow, including laminar benchmark cases and Reτ ∼1,000 turbulence.The turbulent simulations use different channel subdomains, locations, time intervals, and loss-function weights.
- NSFnet formulations: NSFnets use spatial and temporal coordinates as inputs and represent either velocity and pressure or velocity and vorticity outputs under the corresponding Navier–Stokes formulation.The two formulations are called VP-NSFnet and VV-NSFnet.
- Physics-informed construction: Automatic differentiation embeds Navier–Stokes differential operators into the PINN without mesh generation, while the loss function incorporates governing equations and constraints.For VP-NSFnet, pressure is inferred as a hidden state through incompressibility without an additional Poisson pressure equation or pressure data.
- Weighting and turbulence: Dynamic loss weights are studied to accelerate training and enhance accuracy, and the paper reports the first direct PINN simulations of turbulence.The influence of loss-function weights is specifically examined for VP-NSFnet accuracy in turbulent channel flow.
2. Solution Methodology
NSFnets use physics-informed neural networks to solve incompressible Navier–Stokes equations through velocity-pressure (VP) or vorticity-velocity (VV) formulations. Initial and boundary conditions, governing-equation residuals, and tunable loss weights are integrated into training.
- Formulations: Two NSFnets formulate the incompressible Navier–Stokes equations using velocity-pressure and vorticity-velocity variables.The VP network predicts velocity and pressure, while the VV network predicts velocity and vorticity.
- Network design: The networks take spatial and temporal coordinates as inputs and approximate the corresponding flow fields.The VP mapping outputs velocity and pressure; the VV mapping outputs velocity and vorticity.
- Training objective: Physics-informed residuals are computed with automatic differentiation and combined with initial and boundary-condition losses.For VP-NSFnet, residuals include momentum equations and the divergence-free constraint; the loss weights balance components and can accelerate convergence.
- Training objective: Pressure receives no boundary or initial data in VP-NSFnet and is obtained as a hidden state through the incompressibility constraint.This obtains pressure without splitting the Navier–Stokes equations as in traditional CFD methods.
- Loss weighting: Dynamic weights adapt the loss coefficients using back-propagated gradient statistics during training.The coefficients are updated with a moving-average strategy, using λ = 0.1.
- Scope: The study applies both VP and VV NSFnets in numerical experiments, while noting that architecture and optimizer choices could further improve accuracy.The authors state that optimizing network size, learning rate, and optimizer is beyond the scope of the current work.
3. Simulations of laminar flows
NSFnets are evaluated on steady and unsteady incompressible laminar flows using VP and VV formulations, analytical or DNS references, and fixed or dynamic loss weights. Both formulations achieve accurate solutions, while network size, formulation, optimization, and weighting strategy affect performance.
- Simulation cases: The study evaluates NSFnets on 2D steady Kovasznay flow, 2D unsteady cylinder wake, and 3D unsteady Beltrami flow.The experiments compare VP- and VV-NSFnets and investigate dynamic loss weighting.
- Evaluation setup: Reference solutions and initial or boundary data come from analytical solutions or high-fidelity DNS results.Pressure means are shifted to the same value before comparing NSFnet and DNS results.
- Kovasznay flow: Both VP and VV formulations attain high-accuracy Kovasznay solutions, with relative errors in the order of 10^-5 to 10^-3.The reported values are summarized as relative L2 errors of velocity and pressure solutions.
- Kovasznay flow: Increasing network size improves performance: VP-NSFnet is better for small networks, whereas VV-NSFnet is more accurate for large networks.This formulation-dependent crossover is reported for the Kovasznay flow tests.
- Loss weighting: Dynamic weights outperform fixed α = 1 and α = 100 settings for both formulations in the Kovasznay tests.The dynamic coefficient oscillates and varies with changes in learning rate; dynamic weighting also produces more accurate solutions during training.
- Loss weighting: Unweighted physics and boundary-loss gradients are unbalanced, while dynamic weighting makes their gradient histograms more consistent.The preferred dynamic-weight formulation differs between VP- and VV-NSFnets in this assessment.
- Cylinder wake: For the cylinder wake, NSFnet vorticity contours qualitatively agree with DNS, and dynamic weights improve accuracy for both VP and VV formulations.The reported time-dependent comparison is based on relative L2 errors versus time.
- Beltrami flow: For Beltrami flow, both NSFnets obtain accurate Navier-Stokes solutions, with VV-NSFnet outperforming the VP formulation.The simulations use high-fidelity DNS data as a reference and for training boundary and initial data.
4. Simulations of turbulent channel flow
The study applies VP-NSFnet to turbulent channel flow across varying domains and time intervals, finding sustained turbulence and generally accurate velocity predictions, while pressure and crossflow errors remain higher.
- Long-time simulation: VP-NSFnet sustains turbulence over a long time period and obtains good agreement with DNS after loss-function convergence.The long-time test covers more than two local convective time units in one configuration.
- Large-domain simulation: In the larger domain, VP-NSFnet remains very accurate, but pressure relative L2 error can reach 17%.The domain covers half the channel height and includes multiple turbulent regions across the wall-normal direction.
- Large-domain simulation: Wall-normal and spanwise velocity L2 errors are nearly an order higher than streamwise error because streamwise velocity has larger amplitude.The authors suggest careful normalization and anisotropic weight tuning to obtain more balanced accuracy.
5. Summary
NSFnets use PINNs with VP and VV Navier–Stokes formulations to simulate incompressible flows from laminar to turbulent regimes. The simulations achieve good agreement after convergence, while dynamic loss weighting improves accuracy but VV training remains difficult in the reported turbulent setting.
- Summary: NSFnets formulate PINNs using velocity-pressure and velocity-vorticity forms of the incompressible Navier–Stokes equations.The VP-NSFnet outputs velocity and pressure, whereas the VV-NSFnet outputs velocity and vorticity.
- Summary: Spatial and temporal coordinates are inputs, while initial and boundary conditions provide supervised data and equation residuals provide the physics-informed loss component.Automatic differentiation represents the differential operators in the governing equations.
- Summary: Pressure boundary and initial data are not supplied to VP-NSFnet; pressure is obtained as a hidden state through the incompressibility constraint without splitting the equations.The reported setup provides only velocity initial and boundary values for training.
- Summary: Both NSFnets simulate laminar examples, and VP-NSFnet achieves good agreement with DNS after convergence in turbulent channel-flow tests.The study covers Kovasznay flow, cylinder wake, Beltrami flow, and turbulent channel flow.
- Summary: VV-NSFnet is more accurate than VP-NSFnet for laminar cases, while dynamic weights improve the performance of both formulations.The paper also reports enhanced VP-NSFnet accuracy from dynamic weighting in turbulent simulations.
- Summary: VV-NSFnet training fails to achieve satisfactory governing-equation loss convergence in the reported turbulent case, despite reasonable accuracy at α = 50,000.Boundary-condition loss becomes small, but the governing-equation residual remains very large; the authors relate this to potentially inconsistent VV boundary data derived from VP-form DNS data.