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Enhancing Computational Fluid Dynamics with Machine Learning
Ricardo Vinuesa, Steven L. Brunton
TL;DR
Computational fluid dynamics faces costly or impractical simulations of turbulent flows, motivating machine-learning approaches. This Perspective examines ML for accelerating simulations, improving turbulence models, and enhancing reduced-order models while identifying promising opportunities and limitations.
Problem
Turbulent-flow simulations can be computationally expensive or impractical at high Reynolds numbers, while DNS is also hindered by complex geometries.
Method
The Perspective synthesizes machine-learning applications for accelerating simulations, improving turbulence models, developing reduced-order models, and exploring emerging CFD opportunities.
Results
The Perspective identifies high potential for machine learning across simulation acceleration, turbulence modeling, reduced-order modeling, and emerging CFD applications.
Takeaways & Limitations
Machine learning offers promising avenues for advancing CFD, provided models incorporate physical constraints and uncertainties are identified and quantified.
Takeaways & Limitations
ML methods may require expensive training and large datasets, and can be less efficient or straightforward than established classical methods in some CFD applications.
Abstract
from arXiv · showhide
Machine learning is rapidly becoming a core technology for scientific computing, with numerous opportunities to advance the field of computational fluid dynamics. In this Perspective, we highlight some of the areas of highest potential impact, including to accelerate direct numerical simulations, to improve turbulence closure modeling, and to develop enhanced reduced-order models. We also discuss emerging areas of machine learning that are promising for computational fluid dynamics, as well as some potential limitations that should be taken into account.
1 Introduction
Computational fluid dynamics models scientifically and industrially important fluid flows governed by nonlinear Navier–Stokes equations. This Perspective examines how machine learning could improve CFD by accelerating high-fidelity simulations, advancing turbulence models, and extending reduced-order models.
- CFD context: CFD simulates fluid flows governed by nonlinear Navier–Stokes partial differential equations modeling conservation of mass and momentum in Newtonian fluids.Fluid mechanics is important scientifically and across industrial-engineering applications.
- Machine learning context: Machine learning enables algorithms to learn from data without explicit mathematical models and is driving advances across scientific and engineering fields.Deep learning has supported many of the field’s most exciting advances through neural networks.
- Perspective scope: The Perspective explores machine learning to increase high-fidelity simulation speed, develop turbulence models at different fidelity levels, and produce enhanced reduced-order models.These aims extend beyond what classical reduced-order approaches can achieve.
2 Accelerating direct numerical simulations
Machine learning can accelerate direct numerical simulations by improving discretization, solving computationally expensive Poisson equations more efficiently, and reducing the simulated domain while retaining physical properties. These opportunities address DNS demands for fine meshes and accurate methods arising from turbulent flows’ multiscale structure.
- DNS challenges: DNS resolves all flow structures in turbulent, multiscale flows, requiring fine meshes and accurate computational methods to avoid numerical artifacts.The governing Navier–Stokes equations are discretized and integrated with enough degrees of freedom to resolve the flow structures.
- Improving discretization: Machine-learning methods improve DNS discretization, including deep-learning estimates of spatial derivatives on low-resolution grids and learned finite-volume schemes.Bar-Sinai et al. reported that learned spatial derivatives outperformed standard finite-difference methods; Stevens and Colonius improved fifth-order finite-difference schemes for shock-capturing simulations.
- Improving discretization: At one tenth the computational cost, a deep neural network simulated a finite-volume discretization scheme for reactive flows with very good agreement with reference high-resolution data.Errors relative to the reference solution increased with time.
- Accelerating Poisson solves: Machine learning can accelerate Poisson solves by learning mappings from the divergence of an uncorrected velocity field to the resulting pressure field.Poisson equations are frequently used in operator-splitting methods for Navier–Stokes discretization, and solving them is typically the numerical solver’s most computationally expensive step.
- Reducing the computational domain: Reducing computational domains can produce savings by replacing upstream or far-field regions with suitable inflow or boundary conditions, with machine learning helping develop these conditions.The omitted regions do not need to be simulated while retaining physical properties of the system.
3 Improving turbulence models
Machine learning is being used to improve RANS and LES turbulence modeling, including Reynolds-stress prediction, RANS convergence, and subgrid-scale modeling. Effective approaches must account for physical constraints, uncertainty, and the limitations of replacing RANS-closure terms with DNS quantities.
- Motivation: DNS is often impractical at high Reynolds numbers and in complex geometries, so industrial CFD typically uses RANS or coarsely resolved LES with modeled smaller scales.RANS simulates no turbulent scales, whereas LES resolves only the largest scales.
- RANS turbulence models: RANS equations describe mean flow after time averaging, while unresolved velocity fluctuations contribute to the closure problem.The instantaneous velocity is decomposed into mean and fluctuating components.
- RANS turbulence models: Machine learning methods target Reynolds-stress accuracy, with invariant architectures improving performance over traditional linear and nonlinear eddy-viscosity RANS models.Ling et al. embedded Galilean invariance through a multiplicative layer and invariant tensor basis for the predicted anisotropy tensor.
- Model limitations: Physical constraints and uncertainty quantification are important for ML turbulence models, while DNS-derived closure replacements can produce unsatisfactory predictions because of RANS assumptions and ill-conditioning.These limitations make compatibility with the RANS formulation important when incorporating DNS quantities.
- RANS turbulence models: RANS-based CNN acceleration achieved convergence 1.9 to 7.4 times faster than the CFD solver in both laminar and turbulent flows.The method combined CFD-solver iterations with CNN evaluation using the Spalart–Allmaras model in OpenFOAM.
- LES turbulence models: In LES, machine learning supplements unresolved energy on coarse meshes or uses agent-based approaches to stabilize simulations, with supervised models trained on high-fidelity data.Reported applications include flame-surface wrinkling and subgrid-scale prediction in Kraichnan turbulence, where learned models improved on classical algebraic models.
4 Developing reduced-order models
Machine learning is advancing reduced-order models by learning lower-dimensional coordinates and dynamical systems for coherent flow structures. These approaches aim to improve ROM accuracy, efficiency, generalizability, interpretability, and data efficiency while incorporating physical knowledge.
- Reduced-order models provide fast, lower-dimensional, lower-fidelity surrogates by describing the evolution of dominant coherent flow structures.They offer coarse but valuable flow information compared with more expensive CFD techniques.
- Developing a machine-learning ROM requires learning reduced coordinates and identifying a dynamical system for their time evolution.Both stages have recently seen substantial advances, including POD-based coordinate learning.
- Machine learning can improve ROM coordinates beyond POD through deep nonlinear autoencoders that learn manifold representations and may improve compression.Shallow linear autoencoders are closely related to POD, whereas deep nonlinear architectures generalize this approach.
- Neural networks, including LSTM and echo-state networks, and alternative regression methods can learn nonlinear dynamics in established reduced coordinates.The approaches include reservoir computing and other dynamical-systems regression techniques.
- Machine-learning ROMs can combine deep autoencoders with approaches such as SINDy and nonlinear-coordinate DMD to model reduced dynamics.These architectures uncover low-dimensional latent spaces or nonlinear coordinate embeddings.
- Future ROMs should improve accuracy, efficiency, generalizability, interpretability, and data efficiency while embedding physics, stability, symmetries, invariances, and sparsity.The target applications include new initial and boundary conditions, flow configurations, and varying parameters.
5 Emerging possibilities and outlook
The outlook identifies emerging ML opportunities for CFD, including non-intrusive sensing and physics-informed neural networks, while emphasizing grand challenges, methodological caveats, and reproducibility needs. Progress is expected to depend on better data, high-performance computing, and improved understanding of emerging techniques.
- Emerging possibilities: Emerging ML opportunities for CFD include non-intrusive sensing and physics-informed neural networks (PINNs).PINNs use automatic differentiation to calculate partial derivatives and enforce PDE equations through a loss function.
- Grand challenges: Accurate coarse-resolution simulations of unforced three-dimensional wall-bounded turbulent flows remain a grand challenge because near-wall TKE production can be distorted by coarse meshes.At very high Reynolds numbers, outer-layer production also becomes relevant.
- Caveats: ML methods can be expensive to train, require large datasets, and should be assessed against established classical methods and the available training-data information.The passage presents these issues as caveats that may limit ML applicability in some CFD areas.
- Caveats: Embedding flow properties such as incompressibility and periodicity can improve training efficiency and prediction accuracy, while transfer learning is promising.Benchmarking should also account for the cost of generating training data.
- Outlook: Future progress will be driven by high-quality data, high-performance computing, improved facility with emerging techniques, and reproducible research standards.The authors advocate continued community attention to data when developing ML models.