Energy-Conserving Neural Network Closure Model for Long-Time Accurate and Stable LES

TL;DR

The paper tackles the instability and conservation violations often seen with data-driven closures in large-eddy simulation. It introduces an energy-conserving, skew-symmetric neural closure that, together with a dissipative negative-definite term, guarantees stable long-time LES on a structure-preserving discretization using a face-averaging filter. Through 2D turbulence tests (decaying and Kolmogorov flow), the approach consistently outperforms the Smagorinsky closure and exhibits superior stability and spectral fidelity relative to unconstrained neural closures, albeit at higher dissipation. The work demonstrates how embedding physical structure into machine-learned closures yields robust, reliable, and transferable performance, with potential for extending to unstructured grids and more complex flows.

Abstract

Machine learning-based closure models for LES have shown promise in capturing complex turbulence dynamics but often suffer from instabilities and physical inconsistencies. In this work, we develop a novel skew-symmetric neural architecture as closure model that enforces stability while preserving key physical conservation laws. Our approach leverages a discretization that ensures mass, momentum, and energy conservation, along with a face-averaging filter to maintain mass conservation in coarse-grained velocity fields. We compare our model against several conventional data-driven closures (including unconstrained convolutional neural networks), and the physics-based Smagorinsky model. Performance is evaluated on decaying turbulence and Kolmogorov flow for multiple coarse-graining factors. In these test cases we observe that unconstrained machine learning models suffer from numerical instabilities. In contrast, our skew-symmetric model remains stable across all tests, though at the cost of increased dissipation. Despite this trade-off, we demonstrate that our model still outperforms the Smagorinsky model in unseen scenarios. These findings highlight the potential of structure-preserving machine learning closures for reliable long-time LES.

Figures (16)

• Figure 1: Staggered grid discretization of Navier-Stokes equations. The pressure points are indexed with whole numbers, while for the velocity components we have an offset of $\frac{1}{2}$ in the appropriate direction.
• Figure 2: Schematic representation of the face-averaging filter. In this example, a single coarse-grained cell contains nine fine-grid cells. Three fine-grid velocity components in $\mathbf{u}_h$ on each of the coarse cell faces are averaged to obtain the filtered velocity field $\bar{\mathbf{u}}_H$.
• Figure 3: Resolved energy contributions for the true closure term, computed for five decaying turbulence simulations which constitute the training data for the machine learning closure models. The trajectories are presented for different levels of coarse-graining. The reference grid has a resolution of $2048 \times 2048$. See section \ref{['sec:experimental_setup']} for the exact simulation conditions. (Bottom-right) Resolved energy trajectories for one of the simulations.
• Figure 4: (Left) Error for each coarse-graining factor as a function of time for the decaying turbulence test case. (Right) Resolved energy trajectories for each coarse-graining factor. For an overview of the methods, see Table \ref{['tab:models']}. The black vertical line indicates $t=5$. Everything to the right of this line corresponds to extrapolation in time.
• Figure 5: Vorticity fields at each point in time for each of the closure models on a $64 \times 64$ grid. Simulations correspond to the decaying turbulence test case. Blank boxes indicate an unstable simulation.