NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations

By Xiaowei Jin, Shengze Cai, Hui Li, G. Em Karniadakis. 2020

1,247 citations recorded by the source corpus.

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.

Subfields: Aerospace Engineering, Computational Mechanics, Statistical and Nonlinear Physics

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, methodology): “This strategy has been widely used in existing PINN methods [8, 9, 19, 37].”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “Here, we also use the technique of dynamic weights [11,37].”
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2021, methodology): “[13] applied FC-PINNs to solve Navier-Stokes equations, ranging from laminar to turbulent regimes, while Mao et al.”
Characterizing possible failure modes in physics-informed neural networks (2021, background): “This has garnered interest and shown successful results in a wide variety of science and engineering problems and applications [3, 11, 16, 29–31 However, there have also been issues observed with this formulation.”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “PINNs have trouble propagating information from the initial condition or boundary condition to unseen areas of the interior or to future times as an iterative solver (Jin et al, 2021; Dwivedi and Srinivasan, 2020).”
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks (2020, background): “PINNs in particular, have demonstrated remarkable power in applications including ﬂuid dynamics [12, 13, 14], biomedical engineering [15, 16, 17], meta-material design [18, 19], free boundary problems [20], Bayesian networks and uncertainty quantiﬁcation [21, 22], high dimensional PDEs [23, 24,…”
Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks (2021, methodology): “The PINN framework (Raissi et al. 2019) has also been applied to perform complex flow simulations (Sun et al. 2020; Jin et al. 2021).”
Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions (2021, background): “…applied to various problems in engineering and biology with applications including but not limited to heat transfer (Cai et al., 2021), solution of Navier-Stokes equations in fluid mechanics (Jin et al., 2021), high speed fluid flow (Mao et al., 2020) and solid mechanics (Haghighat et al., 2021).”
When and why PINNs fail to train: A neural tangent kernel perspective (2020, background): “Thanks to the approximation capabilities of neural networks, physics-informed neural networks (PINNs) have already led to a series of remarkable results across a range of problems in computational science and engineering, including fluids mechanics [1, 2, 4, 5], bio-engineering [6, 7], meta-material”
CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method (2021, background): “PINNs have been demonstrated for various physics phenomena, including heat transfer [23,24], fluid dynamics [25–27] and electromagnetic propagation [28–30].”