[Paper Review] Data-driven deconvolution for large eddy simulations of Kraichnan turbulence

Romit Maulik, Omer San+2 more|arXiv (Cornell University)|Dec 5, 2018

Fluid Dynamics and Turbulent Flows4 citations

TL;DR

This paper proposes a physics-informed, data-driven large eddy simulation framework using artificial neural networks to learn optimal convolution and deconvolution maps for sub-grid scale modeling in two-dimensional Kraichnan turbulence. The method predicts an effective eddy viscosity without relying on phenomenological models, achieves accurate $k^{-3}$ energy spectrum scaling, and maintains numerical stability through statistical truncation, outperforming traditional Smagorinsky and Leith models with minimal network complexity.

ABSTRACT

In this article, we demonstrate the use of artificial neural networks as optimal maps which are utilized for convolution and deconvolution of coarse-grained fields to account for sub-grid scale turbulence effects. We demonstrate that an effective eddy-viscosity is predicted by our purely data-driven large eddy simulation framework without explicit utilization of phenomenological arguments. In addition, our data-driven framework precludes the knowledge of true sub-grid stress information during the training phase due to its focus on estimating an effective filter and its inverse so that grid-resolved variables may be related to direct numerical simulation data statistically. The proposed predictive framework is also combined with a statistical truncation mechanism for ensuring numerical realizability in an explicit formulation. Through this we seek to unite structural and functional modeling strategies for modeling non-linear partial differential equations using reduced degrees of freedom. Both a priori and a posteriori results are shown for a two-dimensional decaying turbulence case in addition to a detailed description of validation and testing. A hyperparameter sensitivity study also shows that the proposed dual network framework simplifies learning complexity and is viable with exceedingly simple network architectures. Our findings indicate that the proposed framework approximates a robust and stable sub-grid closure which compares favorably to the Smagorinsky and Leith hypotheses for capturing the theoretical $k^{-3}$ scaling in Kraichnan turbulence.

Motivation & Objective

• To develop a data-driven large eddy simulation framework that bypasses traditional phenomenological sub-grid models for turbulence.
• To learn optimal convolution and deconvolution maps via neural networks without requiring true sub-grid stress data during training.
• To ensure numerical realizability and stability through statistical truncation mechanisms that preserve backscatter.
• To unify structural and functional modeling strategies by combining filter estimation with effective eddy viscosity prediction.
• To demonstrate robustness and generalization using both a priori and a posteriori validation on decaying 2D turbulence.

Proposed method

• A dual neural network architecture is trained to learn the inverse relationship between coarse-grained (LES) and direct numerical simulation (DNS) fields, effectively estimating the filter and its inverse.
• The framework uses only grid-resolved variables and DNS data for statistical mapping, avoiding explicit access to sub-grid stresses during training.
• A statistical truncation mechanism is applied to prevent numerical instability by controlling backscatter in the predicted sub-grid stresses.
• The method employs a linear map representation for convolution and deconvolution, ensuring solenoidal constraints are preserved and enabling applicability to higher dimensions.
• Hyperparameter sensitivity analysis is conducted to assess the impact of network complexity on performance, revealing that simple architectures suffice.
• The approach is validated using a priori and a posteriori tests on a 2D decaying turbulence case with energy spectrum analysis.

Experimental results

Research questions

• RQ1 Can a purely data-driven framework learn effective sub-grid closures without explicit knowledge of sub-grid stresses?
• RQ2 Does the proposed neural network-based deconvolution method accurately recover the $k^{-3}$ energy spectrum scaling in 2D Kraichnan turbulence?
• RQ3 Can statistical truncation mechanisms ensure numerical realizability and stability in explicit LES computations?
• RQ4 How does the performance of the dual network framework compare to classical models like Smagorinsky and Leith in terms of accuracy and stability?
• RQ5 To what extent does network complexity affect the predictive performance of the data-driven closure?

Key findings

• The proposed data-driven framework successfully predicts an effective eddy viscosity without relying on phenomenological assumptions or explicit sub-grid stress data.
• The method achieves accurate $k^{-3}$ energy spectrum scaling in 2D decaying turbulence, matching theoretical expectations and outperforming classical models.
• Statistical truncation mechanisms prevent numerical instability, with one formulation showing strong agreement with DNS statistics while the other over-truncates.
• A hyperparameter sensitivity study reveals that increasing network complexity yields no significant improvement in a posteriori performance, indicating that simple architectures are sufficient.
• The linear map formulation ensures solenoidal constraints are preserved, enabling direct application to higher-dimensional flows.
• The framework respects frame-invariance on the specified mesh due to its data-local, stencil-based design, enhancing robustness and generalizability.