Source-linked AI summary
Sub-grid modelling for two-dimensional turbulence using neural networks
Romit Maulik, Omer San, Adil Rasheed, Prakash Vedula
TL;DR
The paper addresses the challenge of modelling unresolved sub-grid effects in LES without relying on phenomenological closure assumptions. It trains ANNs on localized coarse-grid information, including flow stencils and eddy-viscosity kernels, and evaluates them both a-priori and a-posteriori. The framework stabilizes coarse-grained turbulence simulations and shows that deployment-aware model selection improves architectural choice.
Problem
LES requires closure models for unresolved scales, but conventional sub-grid closures use empirical or phenomenological relationships that can provide insufficient or excessive dissipation.
Method
The framework trains ANNs to map localized vorticity and streamfunction stencils plus Smagorinsky and Leith kernel information to the sub-grid source term.
Results
The proposed framework stabilizes coarse-grained flow and preserves inertial-range spectral trends in decaying two-dimensional turbulence, while deeper networks can yield more accurate spectra despite worse training mean-squared error.
Takeaways & Limitations
A-priori and a-posteriori analyses should be coupled when selecting turbulence-closure architectures because deployment constraints and numerical errors affect predictive accuracy.
Takeaways & Limitations
The study is a proof of concept, with future improvements identified through additional flow invariants, physics-informed optimization, symmetry considerations, and outlier preprocessing.
Abstract
from arXiv · showhide
In this investigation, a data-driven turbulence closure framework is introduced and deployed for the sub-grid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized grid-resolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the sub-grid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a-priori and a-posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability density function based validation of sub-grid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying two-dimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for sub-grid quantity inference. In addition, it is also observed that some measure of a-posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data.
1. Introduction
The paper frames LES sub-grid closure as a physics-dependent problem and proposes a data-driven ANN framework for predicting unresolved vorticity forcing in Kraichnan turbulence. It combines a-priori model selection with a-posteriori deployment to assess accuracy and physical realizability.
- Motivation: Because sub-grid physics depends on the flow phenomenon, the paper motivates an equation-free ANN model for the unresolved source term.The framework also seeks to connect model selection with deployment to identify efficient models and informative grid-resolved quantities.
- Approach: The proposed methodology predicts unresolved vorticity forcing or damping and avoids phenomenological closure expressions.The framework is developed for Kraichnan turbulence using combined a-priori and a-posteriori analyses.
- Approach: The study uses extremely sub-sampled data, performs a-priori hyper-parameter optimization, and tests spectral-scaling recovery during a-posteriori flow evolution.The a-posteriori assessment compares the learned framework with robust algebraic eddy-viscosity models.
- Findings: The combined a-priori and a-posteriori workflow supports optimal kinetic-energy-spectrum prediction and hyper-parameter selection across related flows.The paper reports that eddy-viscosity kernels constrain larger-scale performance and that suitable data locality improves accuracy.
- Mathematical setting: In the reduced-order LES, truncated fine wavenumbers generate a sub-grid term that must be approximated by a model.The governing formulation uses vorticity and streamfunction variables linked through a Poisson equation.
- Motivation: LES computes the largest turbulent scales while closures model contributions from unresolved smaller scales.The unresolved-scale effects produce unclosed terms representing local, instantaneous exchanges between resolved and unresolved scales.
2. Machine learning architecture
The model is a multilayer perceptron that maps localized coarse-grid flow information to a predicted sub-grid source term. Its default input combines vorticity and streamfunction stencils with two eddy-viscosity features, while architecture selection uses grid search and cross-validation.
- Architecture: An ANN establishes a map from an input vector in P-dimensional space to an output vector in Q-dimensional space through hidden-layer transformations.Layers apply biasing and activation functions, with neurons representing components of intermediate vectors.
- Input and output spaces: The default model maps 20 inputs to one predicted sub-grid source value at each coarse-grid location.The 20 inputs comprise nine-point vorticity and streamfunction stencils plus Smagorinsky and Leith kernel information.
- Input and output spaces: The eddy-viscosity kernels provide distilled quantities of interest that support simpler architectures and shorter training durations.The paper also describes these kernels as introducing regularization during optimization.
- Prediction processing: Predicted sub-grid forcing is post-processed before insertion into the vorticity equation to ensure numerical stability.The constraint accommodates potentially negative eddy-viscosities while retaining predictions of positive or negative source values.
- Architecture selection: ReLU activations are used in hidden layers to support efficient optimization and avoid vanishing-gradient problems associated with sigmoidal activations.The architecture is selected before training and deployment.
- Architecture selection: Grid-search architecture selection is coupled with three-fold cross-validation across training, testing, and validation partitions.The search varies network depth and then tunes neuron counts with the selected layer count.
3. Training and validation
The framework combines supervised learning from filtered DNS data with hyper-parameter tuning and validation before a-posteriori deployment. Its a-priori tests assess closure predictions, coefficient sensitivity, generalization across Reynolds numbers, and stability-preserving truncation.
- Data and optimization: Filtered DNS fields provide labeled grid-resolved inputs and outputs for supervised learning of the sub-grid source term.The grid-resolved variables are generated using a Fourier cut-off filter applied to fully resolved DNS fields.
- Data and optimization: Training uses two-thirds of the dataset, validation uses the remainder, and similar training and validation losses indicate an optimized map.The final network is selected by the best validation loss after 5000 iterations using mean-squared error and Adam optimization.
- Data and optimization: A two-layer network with 50 neurons is selected through hyper-parameter optimization, while later testing requires a-posteriori examination.The architecture is chosen from layer and neuron searches; the paper distinguishes validation from testing during evolving-flow deployment.
- A-priori assessment: Static Smagorinsky and Leith closures show accuracy that depends non-trivially on their model coefficients in a-priori probability-density comparisons.Ideal reconstructions occur near coefficient values of 1.0, differing from theoretically accepted three-dimensional Smagorinsky values.
- A-priori assessment: The learned framework reconstructs sub-grid contributions for Re = 32000 and unseen Re = 64000 data using only grid-resolved quantities.The Re = 64000 case is described as a stringent out-of-training validation with more energetic physics.
- Numerical realizability: Hardwired truncation removes roughly half of predicted sub-grid terms to preserve viscous stability during coarse-grained forward simulations.Negative sub-grid values represent damping and positive values represent production; the observed truncation proportion is consistent with approximately equal forward and backscatter.
4. Deployment and a-posteriori assessment
A-posteriori deployment tests whether the learned closure stabilizes coarse-grained Kraichnan turbulence and preserves kinetic-energy spectral behavior under varied Reynolds numbers and realizations.
- Deployment and assessment: Angle-averaged kinetic energy spectra assess whether the dynamic closure preserves integral and inertial-range statistics despite spatial and temporal discretization errors.The framework is applied dynamically in space and time, with second-order spatial discretization used in the numerical implementation.
- Deployment and assessment: At Re = 32000, the learned framework stabilizes coarse-grained flow, reduces cutoff-wavenumber noise, and preserves trends consistent with k−3 inertial-range scaling.UNS accumulates high-wavenumber noise, while DNS exhibits the expected k−3 scaling for two-dimensional turbulence.
- Generalization: The same learned map recovers similar spectral trends for Re = 64000, providing a deployment test beyond the Reynolds number used for training.This comparison is used to assess whether the closure is generalizable rather than a numerical artifact.
- Baseline comparisons: Static Smagorinsky and Leith models depend strongly on eddy-viscosity coefficients for lower-wavenumber energy capture and scaling, with coefficients Cs = Cl = 1.0 causing over-dissipation.The learned framework is compared against these algebraic closures and against no-model and filtered-DNS fields.
- Generalization: Across 24 ensemble-averaged simulations with random vorticity fields sharing the same initial energy spectrum, the framework produces repeatable performance at Re = 32000 and Re = 64000.The deployment is also extended to t = 6, beyond the training interval t = 0 to t = 4, with adequate prediction of vorticity forcing.
- Input design: Including eddy-viscosity kernels can constrain a-posteriori statistical fidelity even when their effect is not visible in a-priori error minimization.For the two-layer network, the kernels improve constrained deployment behavior at Re = 32000.
5. A-priori and a-posteriori dichotomy
The study shows that low a-priori training error does not reliably identify the best deployable closure, making a-posteriori testing necessary for architecture and input selection.
- A-priori and a-posteriori dichotomy: A-priori hyper-parameter selection based primarily on mean-squared error can favor architectures more prone to over-fitting and extrapolation.The authors therefore embed a-posteriori prediction into model-selection decisions.
- Eddy-viscosity inputs: Removing Smagorinsky and Leith viscosity inputs leaves training performance nearly unchanged but produces unconstrained large-scale structures during deployment.The 18-dimensional vorticity-and-streamfunction input lacks the implicit regularization associated with the kernel inputs.
- Eddy-viscosity inputs: For a five-layer network, viscosity kernels increase training error while maintaining constrained prediction of k−3 spectral scaling and improving agreement with DNS at large scales.The deeper network with kernels provides better large-scale statistical agreement than the corresponding shallower configuration.
- Architecture selection: The deeper network predicts the Kraichnan turbulence spectrum more accurately in deployment than the two-layer network despite its larger training mean-squared error.Physical realizability constraints and numerical deployment errors can make a-posteriori-optimal architectures differ from a-priori-optimal ones.
- Stencil selection: A five-point stencil has similar a-priori training errors to the nine-point stencil but fails to capture the nonlinear resolved-to-cutoff-scale relationship during deployment.The resulting loss of statistical fidelity is especially evident at lower wavenumbers; deeper networks make the five- and nine-point stencils more similar statistically.
- Overall selection principle: Optimal sub-grid maps require combined a-priori and a-posteriori study because realizability constraints and numerical errors can produce unexpected deployment behavior.The authors recommend lightweight deployment checks before applying selected architectures to representative flows.
6. Conclusions
The study develops a data-driven neural-network closure for decaying two-dimensional turbulence and evaluates it through both a-priori and a-posteriori analyses. The framework provides dynamic sub-grid dissipation, while model selection must account for truncation and forward-simulation errors.
- Conclusions: The framework maps grid-resolved variables and eddy-viscosities to a dynamic closure strength using an artificial neural network trained on extremely sub-sampled direct numerical simulation data.Inputs include sampled stencils of vorticity and streamfunction together with eddy-viscosity information.
- Conclusions: Prediction truncation and numerical errors during forward deployment require a-posteriori analyses when selecting architectures and input spaces.The conclusion specifically identifies hidden-layer count and input spaces as model-selection choices affected by these errors.
- Conclusions: The proposed data-driven spatio-temporally dynamic sub-grid models are intended for tractable cases such as Kraichnan and Kolmogorov turbulence before practical flow deployment.The paper frames this as a prerequisite involving both a-priori and a-posteriori study.
- Conclusions: The work is a proof-of-concept, with future improvements including additional grid-resolved quantities, physics-informed hyper-parameter optimization, symmetry considerations, and outlier preprocessing.The stated longer-term goal is closure maps that classify requirements across inhomogeneous computational domains and multiple flow classes.