Data-driven modeling of multiscale phenomena with applications to fluid turbulence

1. Problem Statement

The paper addresses the fundamental challenge of modeling multiscale phenomena, specifically fluid turbulence, where physical processes occur across length and time scales varying by orders of magnitude.

• The Limitation of Direct Numerical Simulation (DNS): Solving the Navier-Stokes equations directly is computationally intractable for high Reynolds numbers (e.g., atmospheric or oceanic flows) because the number of degrees of freedom scales as $Re^{9/4}$ (in 3D) or requires resolving scales down to microns.
• The Limitation of Current Coarse-Graining Models: To bypass DNS limitations, methods like Large Eddy Simulation (LES) and Reynolds-Averaged Navier-Stokes (RANS) filter out small scales. However, this introduces a closure problem: the effect of unresolved (subgrid) scales on resolved (large) scales must be modeled.
  • Phenomenological Models: Existing models (e.g., Smagorinsky, Dynamic Mixed) rely on empirical assumptions (isotropy, homogeneity) that often fail in the presence of coherent structures.
  • Data-Driven Models: Recent machine learning approaches often produce "black-box" neural networks that lack interpretability or fail to capture specific physical mechanisms like backscatter (energy transfer from small to large scales), which is critical in 2D turbulence and magnetohydrodynamics.
• Core Gap: There is a lack of a general, data-driven framework that can infer explicit, interpretable, and equivariant effective field theories (EFTs) without relying on specific phenomenological assumptions, capable of accurately capturing backscatter.

2. Methodology

The authors propose a novel data-driven framework to infer an EFT from fundamental descriptions (DNS data) using the SPIDER (Sparse Physics-Informed Discovery of Empirical Relations) algorithm.

• Coarse-Graining Strategy:
  • They apply a Gaussian filter (kernel $G_\Delta$) to DNS data to separate scales into resolved ($\bar{u}$) and subgrid ($u'$) components.
  • Unlike standard LES which discards subgrid variables, this framework seeks to identify the governing equations for the subgrid scales to close the system.
• Decomposition of Subgrid Stress:
  • The subgrid stress tensor $\tau_{ij}$ is decomposed into Leonard ($L$), Cross ($C$), and Reynolds ($R$) stresses.
  • Using moment expansion, $L$ and $C$ are shown to scale as $O((\Delta/\ell_c)^2)$ and $O((\Delta/\ell_c)^4)$ respectively, while $R$ scales as $O((\Delta/\ell_c)^6)$.
• Iterative Inference Procedure:
  1. Inhomogeneous Regression: The framework searches for sparse functional relations for $\tau_{ij}$ in terms of resolved fields ($\bar{u}, \bar{p}$).
     • It first identifies the Nonlinear Gradient Model (NGM2): $\tau^{(2)} \propto (\nabla \bar{u})^2$.
     • It then identifies the next-order correction NGM4: $\tau^{(4)} \propto (\nabla^2 \bar{u})^2$.
  2. Residual Analysis: The authors find that even NGM4 fails to accurately predict energy flux (specifically backscatter). The residual $\tau - (\tau^{(2)} + \tau^{(4)})$ cannot be parameterized by resolved fields alone.
  3. Introduction of New Variables: The residual is identified as the Reynolds stress ($R$), which represents the interaction of subgrid scales. $R$ is treated as a new independent tensor field in the EFT.
  4. Homogeneous Regression: An evolution equation for the new variable $R_{ij}$ is inferred using sparse regression on the residual data.
• Symmetry Constraints: The framework enforces equivariance (invariance under translation, rotation, and Galilean transformations) by constructing term libraries based on group representation theory (specifically $SO(2)$ for 2D flows).

3. Key Contributions

• The NGMR Model: The authors derive a new subgrid-scale model called NGMR (Nonlinear Gradient Model + Reynolds stress). It combines:
  • Explicit gradient terms ($\tau^{(2)}$ and $\tau^{(4)}$) derived from resolved fields.
  • An explicit evolution equation for the subgrid Reynolds stress tensor $R_{ij}$.
• Explicit Evolution Equation for Subgrid Scales: The paper derives a closed-form evolution equation for $R_{ij}$:
  $$\partial_t R_{ij} + \bar{u}_k \nabla_k R_{ij} \approx R_{ik} \nabla_k \bar{u}_j + R_{jk} \nabla_k \bar{u}_i + \nu \nabla^2 R_{ij} - \frac{1}{2}|\bar{S}|R_{ij}$$
  This equation captures the advection, production, diffusion, and dissipation of subgrid energy.
• General Framework for Multiscale Systems: The authors generalize this approach to any system with quadratic nonlinearities, proposing a systematic way to distinguish between "resolvable" (parameterizable via moment expansion) and "unresolvable" (requiring new dynamic variables) components of closure terms.
• Interpretability: Unlike neural network closures, the resulting model consists of explicit differential equations with physically meaningful coefficients derived from dimensional analysis and scaling arguments.

4. Results

The framework was tested on 2D incompressible fluid turbulence using Direct Numerical Simulation (DNS) data for three flow regimes: inverse cascade, freely decaying turbulence, and direct cascade.

• Accuracy Metrics: The model was evaluated using magnitude-aware correlation ($C$) for the stress tensor and energy flux, and the ratio of predicted to actual net flux ($q_\Pi$).
• Performance Comparison:
  • Phenomenological Models (DS, DM): Failed to predict local energy fluxes and significantly underestimated net flux (often <30% accuracy). They could not capture backscatter.
  • NGM2/NGM4: While NGM4 achieved high correlation for the stress tensor ($>99\%$), it failed to predict energy flux (0% correlation) because it incorrectly predicted zero net flux.
  • NGMR Model: Achieved >96% correlation for both the stress tensor and energy flux across all test cases. Crucially, it correctly captured backscatter (regions where energy flows from small to large scales), a feature where all other models failed.
• Robustness: The model maintained high accuracy across different Reynolds numbers ($Re \sim 10^4 - 10^7$) and initial conditions. The inferred coefficients were consistent with dimensional analysis (e.g., viscosity $\nu$ appearing in the diffusion term).

5. Significance

• Solving the Backscatter Problem: This is the first data-driven model, to the authors' knowledge, that successfully and explicitly models backscatter in 2D turbulence without ad-hoc corrections (like clipping negative viscosity).
• Paradigm Shift in Modeling: The paper demonstrates that accurate multiscale modeling requires introducing new dynamic variables (tensor fields) for unresolvable scales, rather than just trying to parameterize stress as a function of resolved fields. This challenges the standard RANS/LES approach which often relies on scalar fields (like $k-\epsilon$).
• Efficiency and Generality: The framework automates the discovery of EFTs, reducing the need for decades of iterative phenomenological tuning. It suggests a path toward modeling other complex multiscale systems (e.g., magnetohydrodynamics, active matter) where scale interactions are complex.
• A Priori vs. A Posteriori: While the current results are "a priori" (using filtered DNS data), the authors emphasize that the explicit, equivariant nature of the derived equations is a prerequisite for successful "a posteriori" (numerical) implementation, offering a more stable foundation than black-box ML models.

In summary, the paper presents a rigorous, data-driven methodology that bridges the gap between fundamental physics and effective modeling, successfully deriving an interpretable, high-fidelity subgrid model that outperforms state-of-the-art phenomenological approaches in capturing complex energy transfer mechanisms.