[Paper Review] Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges

Karan Jakhar, Yifei Guan+3 more|arXiv (Cornell University)|Jun 8, 2023

Meteorological Phenomena and Simulations140 references4 citations

TL;DR

This paper investigates the discovery of interpretable, closed-form subgrid-scale (SGS) closures for turbulence using equation discovery from high-fidelity direct numerical simulations. It shows that common algorithms consistently recover the nonlinear gradient model (NGM), which is analytically derivable via Taylor expansion, but reveals that NGM leads to unstable large-eddy simulations due to missing physical mechanisms like backscattering and diffusion, prompting the need for physics-informed loss functions and metrics in future discovery frameworks.

ABSTRACT

There is growing interest in discovering interpretable, closed-form equations for subgrid-scale (SGS) closures/parameterizations of complex processes in Earth systems. Here, we apply a common equation-discovery technique with expansive libraries to learn closures from filtered direct numerical simulations of 2D turbulence and Rayleigh-Bénard convection (RBC). Across common filters (e.g., Gaussian, box), we robustly discover closures of the same form for momentum and heat fluxes. These closures depend on nonlinear combinations of gradients of filtered variables, with constants that are independent of the fluid/flow properties and only depend on filter type/size. We show that these closures are the nonlinear gradient model (NGM), which is derivable analytically using Taylor-series. Indeed, we suggest that with common (physics-free) equation-discovery algorithms, for many common systems/physics, discovered closures are consistent with the leading term of the Taylor-series (except when cutoff filters are used). Like previous studies, we find that large-eddy simulations with NGM closures are unstable, despite significant similarities between the true and NGM-predicted fluxes (correlations $> 0.95$). We identify two shortcomings as reasons for these instabilities: in 2D, NGM produces zero kinetic energy transfer between resolved and subgrid scales, lacking both diffusion and backscattering. In RBC, potential energy backscattering is poorly predicted. Moreover, we show that SGS fluxes diagnosed from data, presumed the ''truth'' for discovery, depend on filtering procedures and are not unique. Accordingly, to learn accurate, stable closures in future work, we propose several ideas around using physics-informed libraries, loss functions, and metrics. These findings are relevant to closure modeling of any multi-scale system.

Motivation & Objective

• To investigate whether equation discovery from high-fidelity data can yield stable, interpretable subgrid-scale (SGS) closures for turbulent flows.
• To identify why commonly discovered closures—despite high pattern correlation with true fluxes—lead to unstable large-eddy simulations.
• To analyze the physical deficiencies of the nonlinear gradient model (NGM), the most frequently discovered closure, in capturing inter-scale energy transfers.
• To propose physics-informed enhancements to equation discovery frameworks, including tailored loss functions, libraries, and metrics, for future stable and accurate SGS modeling.
• To demonstrate that SGS fluxes diagnosed from data are not unique and depend on filtering procedures, challenging the assumption of a single 'truth' for discovery.

Proposed method

• Applies a standard equation discovery algorithm with expansive symbolic libraries to filtered data from 2D forced homogeneous isotropic turbulence (FHIT) and Rayleigh-Bénard convection (RBC) simulations.
• Uses filtered velocity and temperature fields as input features to discover closed-form expressions for momentum and heat fluxes.
• Compares discovered closures to the analytically derived nonlinear gradient model (NGM), derived via Taylor-series expansion of the SGS stress tensor.
• Evaluates the stability of large-eddy simulations (LES) using NGM closures, analyzing inter-scale energy transfer rates (kinetic and potential energy) and enstrophy fluxes.
• Quantifies the mismatch between true SGS fluxes and NGM predictions by computing pattern correlation (>0.95), while assessing physical consistency via energy transfer and backscattering metrics.
• Proposes integrating physics-informed constraints into the discovery process, such as loss functions that penalize missing backscattering or incorrect energy transfer, and sparsity-aware metrics to improve model fidelity.

Experimental results

Research questions

• RQ1 Why do equation discovery methods consistently recover the nonlinear gradient model (NGM) as the subgrid-scale closure, even when it leads to unstable simulations?
• RQ2 To what extent do the discovered closures accurately represent key physical processes like backscattering and diffusion in 2D turbulence and Rayleigh-Bénard convection?
• RQ3 How do filtering procedures affect the diagnosed SGS fluxes, and does this undermine the assumption of a unique 'truth' for closure discovery?
• RQ4 Can standard pattern-matching loss functions in equation discovery reliably produce stable and physically consistent SGS models?
• RQ5 What physics-informed modifications to the discovery framework—such as loss functions, libraries, and metrics—can improve the stability and accuracy of learned closures?

Key findings

• Common equation discovery algorithms robustly recover the nonlinear gradient model (NGM) as the closed-form SGS closure for momentum and heat fluxes across both 2D forced turbulence and Rayleigh-Bénard convection, regardless of fluid properties.
• The NGM closure is analytically derivable via Taylor-series expansion of the SGS stress tensor, confirming its mathematical consistency with known analytical closures.
• Despite high pattern correlation (>0.95) between true and NGM-predicted fluxes, large-eddy simulations using NGM closures are unstable due to zero net kinetic energy transfer between resolved and subgrid scales in 2D.
• In Rayleigh-Bénard convection, the NGM fails to properly capture backscattering of potential energy, leading to unphysical energy transfer dynamics.
• SGS fluxes diagnosed from high-fidelity data are not unique but depend on the filtering procedure, challenging the assumption of a single, universal 'truth' for closure discovery.
• The study identifies that standard pattern-matching loss functions are insufficient for stable closure discovery, and proposes physics-informed alternatives such as energy transfer-aware loss functions and sparsity-aware metrics to improve physical consistency in future work.