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Perspectives on Machine Learning-augmented Reynolds-averaged and Large Eddy Simulation Models of Turbulence
Karthik Duraisamy
TL;DR
Turbulence models need closure for unresolved physics because high-fidelity simulations are often unaffordable. This review surveys ML augmentations of RANS and LES, emphasizing model-consistent training and physics-informed design; it concludes that generalizable models remain challenging and require carefully constrained, assumption-aware development.
Problem
Coarse-graining produces unclosed terms, while direct simulations can remain unaffordable at the spatial and temporal resolutions required for practical flows.
Method
The paper reviews ML representations of closure terms and model discrepancies, model-consistent training, feature-space selection, and physics-informed constraints for RANS and LES.
Results
The review reports that ML-augmented turbulence models have achieved accurate predictions in isolated scenarios, while emphasizing consistency between learning and prediction environments.
Takeaways & Limitations
Generalizable ML-augmented turbulence models require careful characterization of assumptions together with physically and mathematically informed priors and constraints.
Takeaways & Limitations
ML models do not automatically become applicable beyond the scenarios in which they were trained.
Abstract
from arXiv · showhide
This work presents a review and perspectives on recent developments in the use of machine learning (ML) to augment Reynolds-averaged Navier--Stokes (RANS) and Large Eddy Simulation (LES) models of turbulent flows. Different approaches of applying supervised learning to represent unclosed terms, model discrepancies and sub-filter scales are discussed in the context of RANS and LES modeling. Particular emphasis is placed on the impact of the training procedure on the consistency of ML augmentations with the underlying physical model. Techniques to promote model-consistent training, and to avoid the requirement of full fields of direct numerical simulation data are detailed. This is followed by a discussion of physics-informed and mathematical considerations on the choice of the feature space, and imposition of constraints on the ML model. With a view towards developing generalizable ML-augmented RANS and LES models, outstanding challenges are discussed, and perspectives are provided. While the promise of ML-augmented turbulence modeling is clear, and successes have been demonstrated in isolated scenarios, a general consensus of this paper is that truly generalizable models require model-consistent training with careful characterization of underlying assumptions and imposition of physically and mathematically informed priors and constraints to account for the inevitable shortage of data relevant to predictions of interest. Thus, machine learning should be viewed as one tool in the turbulence modeler's toolkit. This modeling endeavor requires multi-disciplinary advances, and thus the target audience for this paper is the fluid mechanics community, as well as the computational science and machine learning communities.
I. INTRODUCTION
This review examines machine-learning augmentation of RANS and LES turbulence models, emphasizing model-consistent training, feature-space design, and physics-informed constraints for reliable models.
- Machine learning is used to augment RANS and LES models by incorporating data more formally into turbulent-flow model development.
- Early approaches learned Reynolds-stress discrepancies or subgrid-scale quantities from DNS data, often by mapping spatial discrepancies into feature spaces.
- Research expanded from isolated RANS applications to LES, multiphase flows, combustion, flow decomposition, equation discovery, and flow control.
- The review organizes prior work around discrepancy representations, training procedures, feature-space selection, and physics-informed constraints.
- Developing reliable and robust ML-augmented turbulence models remains an open challenge requiring attention to model compatibility and generalization across flow regimes.
II. PROBLEM STATEMENT
Turbulence modeling addresses unresolved terms created by coarse-graining high-fidelity fluid equations, seeking closure approximations expressed through modeled coarse-grained variables.
- Direct simulations remain unaffordable for many practical flows because their required spatial and temporal resolution is too high, motivating reduced-fidelity RANS and LES models.
- Coarse-graining decomposes state variables into resolved quantities and unresolved variables, using filtering in LES and ensemble averaging in RANS.
- The coarse-grained equation is exact, but its unclosed quantity cannot be determined solely from coarse-grained variables.
- Turbulence modeling defines closure terms Fm approximately equal to F using modeled coarse-grained variables and secondary variables governed by additional transport equations.
- Closure construction combines physical insight, mathematics, empiricism, and data, yet many approaches have experienced accuracy saturation.
III. REPRESENTATION
ML-augmented turbulence modeling represents unresolved physics either through corrections to modeled closure terms or through predictions of unresolved quantities.
- Research has focused on representing model inadequacy and embedding those representations appropriately within RANS and LES models.
- The reviewed approaches form a prototypical rather than exhaustive set of representations.
- One group of methods models the impact of unresolved physics directly on coarse-grained variables through Fm(˜qm, ˜sm).
- A second group models unresolved quantities themselves rather than their direct impact on coarse-grained variables.
A. Closure term & model inadequacy representations
Reviewed representations range from direct closure corrections to invariant-based, sparse, and neural models for RANS and LES quantities, including subgrid-scale effects.
- RANS representations: RANS approaches learn Reynolds-stress anisotropy, transport-equation corrections, or closure coefficients from features including tensor invariants.
- RANS representations: Tensor-basis neural networks use five objective invariants of modeled strain-rate and vorticity tensors as features, with neural networks predicting expansion coefficients.
- Interpretable representations: Symbolic and sparse regression provide explicit or simplified closure forms, with sparsity regularization eliminating some candidate model terms.
- LES representations: LES methods learn subgrid-scale viscosity, stress tensors, forcing, or reacting-flow quantities from resolved gradients, stresses, fields, and mesh resolution.
- LES representations: Optimal estimation can separate subgrid-scale modeling error into parametric and model-form components, with neural networks addressing each component.
B. Subgrid scale representations
Subgrid-scale representations use learned mappings to reconstruct unresolved contributions or finer-scale fields from coarse resolved quantities. Deconvolution targets sub-filter terms, while super-resolution generates fine fields or spatio-temporal detail.
- Deconvolution: Approximate deconvolution reconstructs sub-filter quantities from filtered fields to approximate subgrid-scale stresses.Neural networks can learn the deconvolution mapping from localized stencils, without assuming the filtering kernel a priori.
- Comparison: Deconvolution and super-resolution differ in target: the former models closure-related sub-filter contributions, whereas the latter reconstructs fine-scale fields.Fine fields may then be used to compute subgrid terms in momentum and scalar equations.
- Super-resolution: Super-resolution maps coarse fields to fine fields, including finer meshes, higher-order solutions, or intermediate time instances.Spatio-temporal variants use coarse fields at two time instances to generate fields at smaller times within the interval.
- Super-resolution: Generative adversarial networks provide expressive variants for super-resolving coarse fluid solutions and particle-scale velocity fields.Particle-resolved velocity generation can be performed locally and applied patchwise across the domain.
- Evaluation: These approaches have mostly been evaluated a priori, although their expressiveness suggests promise for turbulence modeling.The cited super-resolution approaches are described as promising despite limited online evaluation.
IV. TRAINING & CONSISTENCY
The training-and-consistency discussion distinguishes agreement with DNS data from compatibility with the RANS or LES model environment. It formulates the augmented transport equations to make that modeling environment explicit.
- Training and consistency: The section studies how closure models are extracted from data and whether learning and prediction environments are consistent.The central distinction concerns model extraction rather than only the representation of unclosed terms.
- Mathematical formulation: The ML augmentation is represented as δm(˜ηm;w), while the augmented model is expressed through transport equations Ra(˜qm,˜sm,δm(˜ηm;w)) = 0.The formulation covers RANS or LES transport equations and equations for scale-providing variables.
- Mathematical formulation: This compact formulation provides a common notation for examining the model representations and training procedures discussed in the section.The notation is introduced to exemplify the representations developed for the preceding modeling approaches.
A. A priori training
A priori training is natural and non-intrusive, but matching DNS-based inputs does not ensure consistency with the deployed RANS or LES environment. Error accumulation, feature mismatch, model-term balance, and DNS-data requirements constrain its predictive use.
- Advantages and limits: The training approach is non-intrusive and permits direct imposition of physics-based constraints, but model consistency remains unresolved.These advantages coexist with the need to align the learning environment with the model environment.
- Consistency challenges: A priori training establishes consistency with the DNS field but does not guarantee consistency with the RANS or LES environment.The training procedure uses DNS-derived targets and features, whereas prediction uses coarse-grained model features.
- Consistency challenges: Feature mismatch arises because DNS-derived inputs during training differ from coarse-grained model features during prediction.Successful deployment requires the model-predicted features to closely match the DNS-derived features, which is difficult when secondary variables are inaccurate.
- Consistency challenges: 6.0: accumulated online errors can eliminate fine-scale structure in a super-resolution prediction.The embedded network receives a field corrupted by growing prediction error rather than the clean coarse field used offline.
- Consistency challenges: Successful a priori evaluation is neither necessary nor sufficient for successful predictive turbulence models.The section cites the Smagorinsky model and examples where exact or DNS-derived modeled quantities did not ensure satisfactory predictions.
- Data requirements: A priori techniques require full DNS fields, which are unavailable in practical regimes.This data requirement is identified as a major impediment to applying the approach beyond settings with accessible DNS data.
B. Model-consistent training
Model-consistent training embeds the ML augmentation within the modeled environment, using field inversion, model-output data, or coupled optimization. These approaches improve consistency and can reduce reliance on full DNS fields, but introduce inverse-problem and optimization challenges.
- Field inversion: Field inversion estimates spatio-temporal augmentation fields from sparse experimental or DNS data and can support subsequent learning of a generalizable augmentation.Multiple datasets presumed informative of the model discrepancy are used before learning.
- Field inversion and ML: The FI-ML approach trains a generalized augmentation from model outputs rather than coarse-grained DNS data.This combines field inversion with supervised learning while promoting consistency between model outputs and the learning environment.
- Integrated training: Fully coupled training feeds the ML output into the model, compares the resulting model output with data, and updates ML weights in one optimization step.The constrained formulation minimizes a loss subject to the augmented transport equations.
- Integrated training: Integrated training ensures full consistency between learning and prediction environments, and Figure 2 presents its importance.The approach has been applied in both RANS and LES contexts.
- Challenges: Integrated training requires features to be selected before inference, whereas separating FI and ML preserves greater flexibility.This is identified as a drawback of the tightly integrated approach.
- Challenges: Coupled approaches face large inverse problems, non-convex neural-network losses, PDE scale and nonlinearities, and potentially unstable LES adjoints.These challenges can require adjoint-driven optimization and special treatment of chaotic systems.
- Weak coupling: Weakly coupled methods seek model consistency while avoiding complex adjoint-driven machinery, reducing development barriers.Examples include embedded learning, iterative machine learning, CFD-driven machine learning, and closed-loop training.
- Open questions: Convergence properties and loss-function choices remain insufficiently studied, leaving mathematical analysis an open research direction.The discussion identifies rigorous convergence analysis and further study of loss functions as needed.
V. FEATURE SELECTION
Feature selection for ML-augmented turbulence models should balance generalizability, invariance, locality, dimensionality, and the amount and type of available data. Properly selected and normalized features can support predictions across configurations, but limited data can bias selection and reduce model reliability.
- ML regression models are intended to be interpolative, so feature selection should be guided by turbulence-modeling principles and available data.
- Embedding regression models within a suitable physics-based model can yield better predictive properties on unseen geometries and flow conditions.
- Local non-dimensionalization: Local non-dimensionalization, such as ˜Sk/ε, offers greater potential to generalize across configurations than global non-dimensionalization.
- Invariance considerations: Ideal features should satisfy rotational, reflectional, and frame-invariance properties, including the variables used for local non-dimensionalization.
- Local vs non-local features: Local features are desirable for generalizability and implementation, while wall-distance, wall-stress, and pressure-gradient measures can provide relevant information.
- Data considerations: Random-forest regression can rank 47 features to guide reduction, but limited data may bias feature choice and impact model reliability.A model with d features requires enough data to populate a d-dimensional feature space.
VI. CONSTRAINTS
Physics-informed constraints restrict ML-augmented turbulence models to physically and mathematically compatible behaviors. They can act on inputs, outputs, physics-model observables, or Bayesian priors, reducing the search space to physics-consistent manifolds.
- Constraints can reduce the ML model’s search space to lower-dimensional manifolds consistent with the physics.
- Input constraints on the ML model: Input constraints can restrict the feature space ˜ηm to a manifold satisfying invariance properties.
- Output constraints on the ML model: Output constraints can restrict quantities such as Reynolds-stress perturbations to remain compatible with the baseline model.
- Constraints on the outputs of the physics model: Constraints on physics-model observables can impose equality or inequality conditions during model-consistent training.
- Constraints on the outputs of the physics model: In combustion modeling, the output can be constrained to match integrated heat release from experiments or DNS even when local heat release is absent from the loss.
- Constraint satisfaction via priors: Bayesian inference can use physical and mathematical information to enforce priors on model parameters and outputs.
VII. ADDITIONAL CHALLENGES & PERSPECTIVES
Generalizable ML-augmented turbulence models face limits from information loss, incomplete data, discretization and filtering errors, and unresolved model-form uncertainty. The paper advocates model-consistent, physics- and mathematically informed development, while treating ML as one tool whose broader generalization remains unachieved.
- Irrecoverable model discrepancies: Coarse-grained RANS and LES fields cannot uniquely determine instantaneous velocity realizations, so their dynamics are represented only in an average sense.Modeling structures such as single-point and Markovian closures can also introduce errors that ML augmentation cannot address.
- Identifiability: Mean velocity and Reynolds shear stress were highly accurate under a model-consistent anisotropy perturbation, but anisotropy itself was inaccurate.Using the entire Reynolds stress tensor as data could remedy this example, whereas DNS is infeasible and experiments provide sparse measurements in practical configurations.
- Impact of discretization error and filtering: LES augmentation must disentangle numerical errors from modeling errors because resolved scales commonly contain discretization errors and filtering may be implicit.The discrepancy between filtered Navier–Stokes equations and model equations can be difficult to assess, requiring further refinement of model-consistent training.
- Interpretability: Interpretable ML models are desirable for analysis, implementation, reproducibility, and wider use, but interpretability alone does not ensure applicability beyond training scenarios.Symbolic regression is presented as one appealing route to interpretability.
- Perspectives: The paper recommends judicious ML use grounded in physics constraints, mathematical rigor, and explicit understanding of information available in the data.Quantifying assumptions as distributions and accounting for them in inverse and forward modeling may add robustness, while model-form uncertainty remains open.
- Perspectives: ML-augmented models have improved predictions on training or related problems, but the goal of more generalizable models has not been achieved.Further progress is tied to standards, benchmarks, collaboration, accessibility, and reproducibility.