NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations (2003.06496v1)

Abstract: 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.

• The paper introduces NSFnets that leverage PINNs to solve incompressible Navier-Stokes equations using both velocity-pressure and vorticity-velocity formulations.
• It utilizes automatic differentiation to incorporate initial and boundary conditions into the loss function, thereby bypassing traditional mesh generation.
• NSFnets achieve accurate simulations in laminar flows and turbulent conditions (Reτ ~1,000), with dynamic weight tuning enhancing performance.

Overview of NSFnets: Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

The paper "NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations" explores the development of Physics-Informed Neural Networks (PINNs) designed to effectively simulate incompressible fluid flow, including both laminar and turbulent regime transitions, via the Navier-Stokes equations. The researchers investigated two distinct formulations of these equations: the velocity-pressure (VP) and vorticity-velocity (VV) approaches, leveraging these variants to create NSFnet models within the artificial intelligence framework.

Approach and Methodology

NSFnets use spatial and temporal coordinates as inputs while delivering either instantaneous velocity and pressure fields (for VP-NSFnets) or velocity and vorticity fields (for VV-NSFnets) as outputs. The authors emphasize using the universal approximator property of neural networks backed by automatic differentiation to sidestep conventional mesh generation in fluid dynamics computational models. This advantage is specifically noted in how it aids the simulation of multiphysics problems without the classical numerical complications like artificial dispersion and diffusion errors.

The construction of the loss function within these models includes the initial and boundary conditions as supervised aspects, while the actual differential Navier-Stokes equations are handled as unsupervised steady-state equations, with automatic differentiation replacing direct numerical methods. This insulates the computation from noise and exploits the differentiability of neural network functions, theoretically increasing the robustness of the solution structure.

Results and Analysis

Evaluating against analytical solutions and direct numerical simulation (DNS) benchmarks, NSFnet results demonstrate impressive accuracy for simulating complex incompressible flows across the laminar to turbulent transitions. Key findings reveal that for laminar flow profiles, VV formulations yield superior accuracy relative to VP configurations. However, for computations at higher Reynolds numbers representative of turbulent conditions, VP-NSFnets were successfully able to sustain turbulence using restricted domains, highlighted at $\rm Re_\tau \sim 1,000$, although necessitating velocity boundary conditions sourced from DNS databases.

Moreover, the paper introduces a systematic paper on tuning the weights in the loss function used for data and physics components of NSFnet models, proposing a new computation method for weights. This dynamic weight tuning stands out as a promising methodology to refine simulation efficiencies, achieving accuracy improvements via adjusted weighting schemas traced across simulations.

Implications and Future Directions

The findings highlight that the NSFnet approach, particularly with advanced weight tuning and PINN frameworks, presents a viable alternative for classic fluid dynamic simulations. Such methods can model turbulent fluid dynamics without the grid dependencies typical of conventional methods.

Future avenues speculated within this paper suggest further work in optimizing the computational efficiency of PINNs, perhaps targeting sophisticated multi-node GPU implementations along with dynamic neural architecture adjustments to further improve training artifacts and broader application potentials. This work sets the stage for physicists and computational scientists to explore these methodologies in high-stakes simulations involving more complex boundary and initial conditions beyond current CFD capabilities.

In conclusion, NSFnet's ability to encapsulate complex flow simulation within a neural network framework presents profound implications for computational fluid dynamics and reinforces the potential for AI models as significant tools in scientific computing. The ongoing refinement of these models could drive breakthroughs not only in fluid mechanics but across the wider spectrum of applied physics simulations.