A Perspective on Machine Learning Methods in Turbulence Modelling

1. Introduction

• The review offers a concise, non-specialist perspective on data-driven closure modeling, focusing chiefly on LES and on methods’ challenges, capabilities, and opportunities.It aims to complement existing reviews rather than catalogue specific examples or rank methods, since accepted best practices have not yet converged.Source: §1 · p. 2

1.1. Turbulence and turbulence modeling

• Coarse-scale RANS and LES reduce the cost of resolving turbulence, but filtering creates a closure term that must be modeled using coarse-scale quantities.In LES, the closure depends on the chosen filter; grid-filtered formulations also incorporate discretization effects and errors, so model development requires data-consistency.Source: §1.1 · pp. 4, 5, 7

1.2. Turbulence modeling and machine learning

• ML is a plausible aid because coarse-grid quantities correlate with subgrid terms, while no known analytical closure exists and simple models can fail.ML can learn both closure parameters and functional forms from data, but training and inference data must remain consistent for useful generalization.Source: §1.2 · p. 9

1.2.1. Challenges for ML-augmented turbulence modeling

• Training ML models requires substantial data or cheap data generation, yet experiments and DNS are costly and existing datasets raise consistency concerns.The expense of generating training data should be justified by the resulting model improvement.Source: §1.2.1 · p. 10
• Training and inference must be consistent and robust to distribution shifts, since noisy data, error accumulation, or changed discretization can make predictions unreliable.Even reasonable parameter predictions may be applied to flow quantities unlike those seen during training.Source: §1.2.1 · pp. 10–11
• Closure optimization must consider the full equation system and interactions among terms; physical constraints can improve robustness, accuracy, and training efficiency.Constraints may be encoded through sample selection or loss functions, but these do not guarantee constraint satisfaction at inference; guaranteed properties and the best constraints remain open challenges. Models respecting overall physical balances have outperformed single-contribution approaches in turbulent boundary layers.Source: §1.2.1 · p. 11
• ML turbulence models need confidence estimates, fallback behavior and generalization across Reynolds numbers and modeling limits, while prediction convergence and interpretability remain open challenges.The suggested limits include turning off in laminar flow and at DNS resolution; the authors also argue that understanding ML decisions is necessary for successful models.Source: §1.2.1 · pp. 11–12
• Practical deployment must reconcile GPU-oriented ML software with legacy CPU-based LES and RANS solvers, and the required integration depends on how much of the modeling ML performs.Lightweight local algorithms could be added to existing codes, whereas predicting full flow fields or model terms may require solvers designed around ML; choosing hardware and balancing workloads remain open research questions.Source: §1.2.1 · p. 12
• Model adoption also depends on simplicity, robustness and computational cost, since computational-graph models can be cumbersome for non-specialists to experiment with.The authors note that ML methods can be accurate, while these other aspects remain open for debate.Source: §1.2.1 · pp. 12–13

1.2.2. Opportunities for ML-augmented turbulence modeling

• ML can make external data integral to model development and discover high-dimensional correlations, complementing rather than replacing physical reasoning and potentially expanding the functional basis for closures.This offers a route beyond tuning existing closure constants, such as eddy-viscosity models, toward identifying new correlations from data.Source: §1.2.2 · pp. 13–14
• ML may represent non-stationary, non-homogeneous regime changes and incorporate discretization effects that make LES closure terms specific to a numerical scheme.The authors identify potential applications to intermittency and extreme events; they argue that ML could conceptually include the nonlinear interaction between closure models and discretization.Source: §1.2.2 · pp. 13–14
• Existing numerical and experimental flow data could reduce costly training-data generation, while ML can naturally combine sparse spatial and temporal inputs for turbulence’s non-local behavior.Mining and consistently cleaning existing data remains a challenge, but flexible inputs and nonlinear feature combination may reduce training-data requirements.Source: §1.2.2 · p. 14

• Machine learning is a prominent AI discipline that learns tasks from data; unlike symbolic GOFAI systems [17], it can be applied where sufficient domain knowledge is unavailable.Source: §2 · p. 15
• Algorithms can transfer across structurally similar tasks; a network trained to detect image edges can be adapted to recognize shock waves in flow data.Source: §2 · p. 15
• Supervised learning infers an input–output relationship from labeled samples, while unsupervised learning finds patterns or low-dimensional representations without labeled outputs.Source: §2 · p. 15
• Reinforcement learning learns through interaction with an environment rather than relying on a dataset.Source: §2 · p. 15
• Learning-paradigm boundaries and the assignment of methods can be ambiguous, and some methods form small subcategories outside the usual segmentation.Source: §2 · p. 15
• The discussion emphasizes supervised learning with neural networks, described as state of the art for many applications, including turbulence modeling.Source: §2 · p. 16

2.1. Supervised learning

• Supervised learning fits an input-output mapping from labeled samples and evaluates generalization on unseen data, using regression for continuous outputs or classification for discrete classes.Training adapts model parameters to minimize a loss function; inference uses the trained model to make predictions.Source: §2.1 · p. 16

2.1.1. Random forests

• Random forests train trees on random data or feature subsets and combine their predictions by majority vote to improve generalization over single decision trees.For regression, the ensemble averages individual predictions; Wang et al. used DNS data to correct RANS Reynolds stresses.Source: §2.1.1 · p. 17, Figure 4 · p. 18

2.1.2. Support vector machines

• SVMs separate classes with a hyperplane, using a kernel to compute implicitly in a mapped feature space when the original data are not linearly separable.SVMs can also be used for regression and clustering, and kernel choice can enforce constraints directly.Source: §2.1.2 · pp. 17–18

2.1.3. Artificial neural networks

• ANNs learn weights and biases in nested linear and nonlinear functions; CNNs suit contextual multidimensional data, while RNNs handle ordered sequences through carried state.Residual networks add skip connections to help train very deep CNNs; gating mechanisms in GRU and LSTM variants address standard RNN gradient problems on long sequences [10, 21].Source: §2.1.3 · pp. 18–20

2.2. Unsupervised learning

• Unsupervised learning uses unlabeled data; its highlighted tasks are similarity-based clustering and low-dimensional representation with minimized information loss, while density estimation is outside scope.Clustering can reveal patterns and correlations in datasets.Source: §2.2 · pp. 20–21

2.2.1. k-means

• K-means assigns each point to its nearest centroid, repeatedly updates centroids from assigned points, and stops when clusters cease changing.The method minimizes average variance within clusters; k-means++ is cited as a variant that improves accuracy and offers performance guarantees and estimates.Source: §2.2.1 · pp. 21–22

2.2.2. Autoencoder

• Autoencoders compress inputs into a low-dimensional latent representation and decode it back, with reconstruction error used to assess accuracy.They are conceptually linked to POD, PCA, and SVD; variational autoencoders learn a distribution and have been used to generate synthetic turbulence [14] [40].Source: §2.2.2 · p. 22

2.3. Reinforcement learning

• In reinforcement learning, an agent interacts with a stochastic environment, receives rewards, and adapts a policy to maximize potential future reward across episodes.The policy determines actions given the current state.Source: §2.3 · p. 22
• Q-learning stores expected rewards for state-action combinations in a table and chooses the action promising the highest reward for a current state [72].The approach becomes impractical for large problems because of memory demands and the time needed to explore possible combinations.Source: §2.3 · p. 22
• Deep reinforcement learning uses deep ANNs to learn an RL policy, an approach also used by AlphaGo, the first AI to beat a professional Go player.The paper contrasts this with tabular Q-learning, whose memory and exploration costs are prohibitive for large problems.Source: §2.3 · p. 22

3. Examples of ML-augmented Turbulence modeling

• The review distinguishes parameter estimation with a posited model form, direct prediction of closure terms, and ML approximation of the full turbulent equations.The third approach is motivated partly by being mesh-free and able to work with sparse data.Source: §3 · p. 23

3.1. Parameter estimation

• An LES eddy-viscosity learning example reported a computational-time saving by a factor of 2 to 8 compared with the dynamic Smagorinsky model [44].The work reported good a priori and a posteriori LES results.Source: §3.1 · p. 24

3.2. Closure term estimation

• Direct closure estimation avoids positing a model form and needs only filtered DNS targets, but reduces interpretability and can lack long-term stability.The passage identifies inference-time data-model inconsistency and high-wavenumber error growth as sources of instability.Source: §3.2 · p. 25
• 99.9% cross correlation across investigated filters was reported for GRU predictions [26], while a spatial MLP achieved over 90% a priori correlation with true subgrid force.The spatial MLP approach outperformed classical closures a posteriori when paired with dissipative regularization.Source: §3.2 · pp. 25–26
• Accurate direct closure predictions can still yield diverging long-run LES without stabilization, while generalizability and transfer to practical models remain open questions.Stability training was reported to improve closure-model stability and extend useful predictions.Source: §3.2 · p. 26

3.3. Full PDE modeling

• PINNs approximate a solution field with a differentiable network, then use its derivatives to construct an optimizable PDE residual without labeled data for that residual.Training balances the solution fit and PDE fulfillment.Source: §3.3 · p. 27
• In a turbulent-flow PINN study, short-term predictions agreed well but errors accumulated over time; the divergence constraint also yielded an expected pressure field without pressure training data [23].The reported turbulent predictions had considerably larger approximation errors than the laminar predictions [23].Source: §3.3 · p. 28
• PINNs remain approximate PDE solutions, so conservation is only approximated and their stability, convergence, generalization, and efficiency require further study.The paper also notes that PINNs relax properties relied upon in classical PDE solvers.Source: §3.3 · p. 29

4. Conclusion and outlook

• Successful ML-augmented modeling requires careful attention to the equations, their discretizations, and the conditions in which turbulence models were derived.The authors stress that ML does not make physical reasoning or mathematical rigor obsolete, or model development easier or fool-proof.Source: §4 · p. 29
• Practical grid-filtered LES has multiple Smagorinsky variants whose interactions with discretization-dependent data contribute to large variations in benchmark results.ML can bring these model-data inconsistencies into sharper focus.Source: §4 · p. 30
• ML offers complementary capabilities in feature extraction and identifying correlations, while generalization, interpretability, transfer, and cost-benefit remain open challenges.The authors call for consolidating research directions and generalizing models for real applications.Source: §4 · p. 30
• The authors propose a community benchmark with a well-described, public training dataset—likely from DNS—and a private test set for blind model evaluation.They recommend approaching setup and administration of the validation case as a community effort.Source: §4 · p. 31