Physics-informed data-driven inference of an interpretable equivariant LES model of incompressible fluid turbulence

This paper presents a physics-informed, data-driven, and parameter-free symbolic subgrid-scale model for incompressible fluid turbulence that utilizes a rank-two tensor field to accurately predict energy and enstrophy fluxes across diverse coherent structures, outperforming leading LES models without relying on restrictive phenomenological assumptions.

The Gist

Imagine you are trying to predict the weather. To do this perfectly, you would need to track every single air molecule in the atmosphere. But there are trillions of them, and even the fastest supercomputers can't handle that much data. So, meteorologists use a trick: they divide the sky into a giant grid of boxes. They calculate the wind and temperature for the center of each box, but they ignore the tiny swirls and gusts happening inside the box.

The problem? Those tiny, ignored swirls actually push and pull on the big weather patterns. If you ignore them, your forecast goes wrong. In fluid dynamics, this is called the Subgrid-Scale (SGS) problem. We need a way to guess what those tiny, invisible swirls are doing so our big-picture model stays accurate.

For decades, scientists have tried to guess these swirls using "rules of thumb" (phenomenological assumptions). They say things like, "The tiny swirls probably act like a thick syrup slowing things down." But nature is messy. Sometimes those swirls don't act like syrup; sometimes they act like a chaotic dance that actually adds energy back into the system. The old rules fail when the flow gets complicated, like when giant whirlpools form.

The New Approach: Letting the Data Speak

This paper introduces a new way to solve this problem. Instead of guessing the rules, the authors used a "smart detective" (an AI called SPIDER) to look at millions of high-resolution computer simulations of fluid flow. They asked the AI: "Based on what you see, what is the exact mathematical formula that describes how the tiny swirls affect the big flow?"

Here is how they did it, using some simple analogies:

1. The "Three-Layer Cake" of Turbulence

The authors realized that the "missing" effects (the subgrid scales) aren't just one thing. They are actually three different types of interactions mixed together:

• The Big Swirls interacting with other Big Swirls (Leonard term): Like two large ocean currents bumping into each other.
• The Big Swirls interacting with the Tiny Swirls (Cross term): Like a giant wave crashing over a small ripple.
• The Tiny Swirls interacting with other Tiny Swirls (Reynolds term): Like a chaotic crowd of people jostling each other in a small room.

Old models tried to describe all three with a single, simple formula (like saying "everything is just thick syrup"). This paper says, "No, that's too simple." They built a model that treats the Tiny-Tiny interactions as a separate character in the story.

2. The "Ghost Character" (The New Variable)

In their new model, they introduced a "Ghost Character" (a mathematical field called $R$) that represents the energy and stress of those tiny, invisible swirls.

• Old Models: Just looked at the big picture and guessed the missing parts.
• New Model: It keeps a running diary of the "Ghost Character." It has a specific equation that tells the Ghost Character how to move, spin, and change over time, just like the main characters (the big wind and water).

This is a bit like a movie director. An old model might just say, "The background crowd looks blurry." The new model actually writes a script for the background crowd, telling them exactly how to move so they don't accidentally ruin the scene.

3. The "Symmetry" Rule

The AI was taught a very important rule: Symmetry.
Imagine you take a photo of a swirling fluid and rotate the photo 90 degrees. The physics shouldn't change; the fluid doesn't care which way is "up."
Many AI models break this rule. They might predict the fluid spins one way if you rotate the camera, and the other way if you don't. The authors forced their AI to only find formulas that respect this symmetry. This made the model interpretable (we can read the math and understand it) and stable (it won't crash when you run it).

The Results: Why It Matters

The authors tested their new model against the old "rules of thumb" and other fancy AI models.

• Accuracy: The new model predicted the flow with near-perfect accuracy, even when the grid was very coarse (when we were ignoring a lot of tiny details).
• Backscatter: This is a fancy word for when tiny swirls push energy back into the big swirls (like a small wave pushing a giant boat forward). Old models usually get this wrong or ignore it. The new model got it right.
• Stability: Because the model respects the laws of physics (symmetry), it didn't blow up or produce nonsense numbers when run for a long time.

The Bottom Line

Think of fluid turbulence like a chaotic dance party.

• Old Models tried to describe the party by saying, "Everyone is just moving randomly in a circle."
• This New Model realized that the dance has structure. It identified that the dancers have different roles (big leaders, small followers, and the chaotic crowd in the middle). It wrote a script for the chaotic crowd so they could dance in sync with the leaders.

The result is a model that is simpler (no adjustable knobs to tweak), faster (runs efficiently), and more accurate than anything we had before. It proves that if we let data guide us while respecting the fundamental laws of physics, we can build a much better understanding of how fluids move, from weather patterns to blood flow in our veins.

Technical Summary

1. Problem Statement

Large Eddy Simulation (LES) aims to model turbulent flows by resolving large scales and modeling the effect of unresolved subgrid scales (SGS) via a closure term (the SGS stress tensor, $\tau_{ij}$). Current phenomenological models (e.g., Smagorinsky, Dynamic Mixed) rely on restrictive assumptions such as homogeneity, isotropy, and scale invariance. These assumptions often fail in flows with coherent structures (common in 2D turbulence) and struggle to accurately capture:

• Local energy fluxes and backscatter: The transfer of energy from small to large scales.
• Generalizability: Performance often degrades significantly as the filter scale increases relative to the flow's integral scale.
• Interpretability: Deep learning approaches, while powerful, often lack physical interpretability and stability.

The core challenge is to develop a subgrid model that is parameter-free, interpretable, equivariant (respects physical symmetries), and capable of accurately predicting both stress tensors and local fluxes across a wide range of filter scales and Reynolds numbers.

2. Methodology

The authors employ a hybrid approach combining formal fluid dynamics analysis with the SPIDER (Symbolic Physics-Informed Data-driven Equation Discovery) framework.

A. Theoretical Foundation: LCR Decomposition

The authors decompose the SGS stress tensor $\tau_{ij}$ into three Galilean-invariant components (Leonard, Cross, and Reynolds):
$$\tau_{ij} = L_{ij} + C_{ij} + R_{ij}$$

• $L_{ij}$ (Leonard): Large-large scale interactions.
• $C_{ij}$ (Cross): Large-small scale interactions.
• $R_{ij}$ (Reynolds): Small-small scale interactions.

Using a moment expansion for a Gaussian filter, they establish that $L_{ij} \sim O(\delta^2)$, $C_{ij} \sim O(\delta^4)$, and $R_{ij} \sim O(\delta^6)$, where $\delta = \Delta/\ell_i$ is the ratio of filter scale to integral scale. While standard gradient models (NGM, NGM4) approximate $L$ and $C$, they fail to model $R$ effectively, leading to inaccuracies in flux prediction.

B. Data-Driven Inference (SPIDER)

The authors use SPIDER to infer governing equations directly from Direct Numerical Simulation (DNS) data:

1. Symmetry Enforcement: The algorithm generates term libraries based on irreducible representations of the rotation and Galilean groups, ensuring the inferred model is equivariant.
2. Weak Formulation: Derivatives are evaluated using weak formulations to minimize noise and discretization errors.
3. Sparse Regression: A greedy regression algorithm identifies the most parsimonious set of terms that describe the data.
4. Variable Discovery: Instead of assuming $\tau_{ij}$ depends only on resolved velocity $\bar{u}$, the method introduces a new tensor variable, $R_{ij}$, to represent the subgrid Reynolds stress.

C. The Inferred Model (NGMR)

The resulting model, NGMR, consists of:

1. SGS Stress Tensor:
   $$\tau_{ij} = \underbrace{\frac{\Delta^2}{12}(\nabla_k \bar{u}_i)(\nabla_k \bar{u}_j)}_{L \text{ (NGM)}} + \underbrace{\frac{\Delta^4}{288}(\nabla_k \nabla_m \bar{u}_i)(\nabla_k \nabla_m \bar{u}_j)}_{C \text{ (NGM4)}} + \underbrace{R_{ij}}_{\text{New Variable}}$$
2. Evolution Equation for $R_{ij}$:
   A transport equation for the Reynolds stress tensor is inferred:
   $$\partial_t R_{ij} + \bar{u}_k \nabla_k R_{ij} = (R_{ik}\nabla_k \bar{u}_j + R_{jk}\nabla_k \bar{u}_i) + \nu \nabla^2 R_{ij} - \alpha I R_{ij} + \dots$$
   where $I = \sqrt{I_1 + I_2}$ is a scalar invariant of the strain and rotation tensors. The coefficients are determined via dimensional analysis and scaling (e.g., $\alpha \approx 0.1\text{--}0.25$).
3. Regularization: Hyperviscous terms are added to ensure numerical stability without altering large-scale dynamics.

3. Key Contributions

• Interpretability without Phenomenology: The model is derived entirely from data and symmetry principles, requiring no adjustable parameters or ad-hoc assumptions (like the Boussinesq hypothesis).
• Rank-Two Tensor Subgrid Variable: The paper demonstrates that a scalar subgrid variable (common in RANS) is insufficient. A rank-two tensor field ($R_{ij}$) is required to correctly capture the orientation of small-scale structures (vorticity filaments) relative to large-scale strain, which is crucial for predicting backscatter.
• Hybrid Structure: While inferred via LES-style coarse-graining, the model's structure (an evolution equation for a subgrid tensor) is more akin to Reynolds-Averaged Navier-Stokes (RANS) full-stress models, bridging the gap between the two paradigms.
• Robustness: The model remains accurate for filter scales up to the integral scale ($\delta \approx 1$), a regime where traditional models fail.

4. Results

The model was validated on 2D incompressible turbulence (forced and freely decaying) across various Reynolds numbers ($Re \sim 10^5 - 10^6$) and filter scales.

• A Priori Benchmarks (Static Comparison):

  • Stress Tensor: NGMR achieves near-perfect correlation with DNS for $\tau_{ij}$ across all filter scales.
  • Local Energy Flux ($\Pi$): Unlike phenomenological models (Smagorinsky, Similarity) which fail to predict local fluxes, NGMR accurately captures both forward and backward energy transfer.
  • Net Energy Flux: NGMR is the only model to consistently predict the mean energy flux ratio $\langle \Pi_{LES} \rangle / \langle \Pi_{DNS} \rangle \approx 1$ across all scales.

• A Posteriori Benchmarks (Dynamic Simulation):

  • Energy & Enstrophy Decay: In freely decaying turbulence, NGMR tracks the decay of energy and enstrophy significantly better than Dynamic Smagorinsky, Dynamic Mixed, Similarity, and NGM4.
  • Backscatter: NGMR is the only model that correctly predicts the vanishing of backscatter as vortices merge in the late stages of decay.
  • Statistical Distributions: For forced turbulence, NGMR accurately reproduces the Probability Distribution Functions (PDFs) of vorticity and energy flux, including the tails (extreme events) and the asymmetry between dissipation and backscatter.
  • Short-term Forecasting: NGMR maintains high spatial correlation with DNS for significantly longer times ($t/T_e \approx 250$) compared to other models, which decorrelate rapidly.

• Computational Cost: The computational cost of NGMR is comparable to the Similarity model and roughly half that of Dynamic Smagorinsky/Mixed models, making it efficient despite the additional evolution equation.

5. Significance

This work represents a paradigm shift in turbulence modeling:

1. Validation of Data-Driven Discovery: It proves that symbolic regression can discover complex, stable, and interpretable physical laws that outperform decades of phenomenological modeling.
2. Necessity of Tensor Subgrid Variables: It establishes that for flows with strong coherent structures and backscatter, subgrid models must include tensorial variables to capture directional fluxes, challenging the dominance of scalar-based RANS and eddy-viscosity LES models.
3. Generalizability: The framework is not limited to 2D; the authors argue it generalizes to 3D and compressible flows, offering a blueprint for next-generation turbulence models that are both accurate and physically transparent.
4. Stability and Accuracy: By enforcing equivariance and using weak formulations, the model achieves a level of numerical stability and accuracy previously unattainable without heavy regularization or clipping.

In conclusion, the NGMR model successfully bridges the gap between the interpretability of RANS and the accuracy of LES, providing a robust, parameter-free solution for simulating complex turbulent flows.