# Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges arXiv: https://arxiv.org/abs/2306.05014 ยท HTML: https://mlsanity.com/paper/2306.05014/ ## Key facts - Cited by 9 papers in the map (data through 14 Oct 2026). - Builds on 21 of its 182 references that are in the map. - First posted on arXiv 8 Jun 2023. ## Builds on (most cited first, 21 of 21) - [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019) - [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021) - [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019) - [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019) - [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020) - [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020) - [Deep learning to represent sub-grid processes in climate models](https://mlsanity.com/paper/1806.04731/) (2018) - [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023) - [Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High-Resolution Simulations](https://mlsanity.com/paper/1709.00037/) (2017) - [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021) - [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021) - [A posteriori learning for quasi-geostrophic turbulence parametrization](https://mlsanity.com/paper/2204.03911/) (2022) - [Investigating the fidelity of explainable artificial intelligence methods for applications of convolutional neural networks in geoscience](https://mlsanity.com/paper/2202.03407/) (2022) - [Explaining the physics of transfer learning a data-driven subgrid-scale closure to a different turbulent flow](https://mlsanity.com/paper/2206.03198/) (2022) - [Climbing down Charney's ladder: Machine Learning and the post-Dennard era of computational climate science](https://mlsanity.com/paper/2005.11862/) (2020) - [Data-driven super-parameterization using deep learning: Experimentation with multi-scale Lorenz 96 systems and transfer-learning](https://mlsanity.com/paper/2002.11167/) (2020) - [Explainable Artificial Intelligence for Bayesian Neural Networks: Towards trustworthy predictions of ocean dynamics](https://mlsanity.com/paper/2205.00202/) (2022) - [Direct data-driven forecast of local turbulent heat flux in Rayleigh-B'enard convection](https://mlsanity.com/paper/2202.13129/) (2022) - [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021) - [A machine learning framework for LES closure terms](https://mlsanity.com/paper/2010.03030/) (2020) - [Ensemble Kalman Inversion for Sparse Learning of Dynamical Systems from Time-Averaged Data](https://mlsanity.com/paper/2007.06175/) (2020) ## Built on by (most cited first, 9 of 9) - [Hierarchical Implicit Neural Emulators](https://mlsanity.com/paper/2506.04528/) (2025) - [Online Calibration of Deep Learning Sub-Models for Hybrid Numerical Modeling Systems](https://mlsanity.com/paper/2311.10665/) (2023) - [Deep Learning of Solver-Aware Turbulence Closures from Nudged LES Dynamics](https://mlsanity.com/paper/2604.23874/) (2026) - [Scalable and Consistent Graph Neural Networks for Distributed Mesh-based Data-driven Modeling](https://mlsanity.com/paper/2410.01657/) (2024) - [Guided Unconditional and Conditional Generative Models for Super-Resolution and Inference of Quasi-Geostrophic Turbulence](https://mlsanity.com/paper/2507.00719/) (2025) - [Fourier analysis of the physics of transfer learning for data-driven subgrid-scale models of ocean turbulence](https://mlsanity.com/paper/2504.15487/) (2025) - [An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulence](https://mlsanity.com/paper/2509.20365/) (2025) - [On the importance of learning non-local dynamics for stable data-driven climate modeling: A 1D gravity wave-QBO testbed](https://mlsanity.com/paper/2407.05224/) (2024) - [Global Estimation of Subsurface Eddy Kinetic Energy of Mesoscale Eddies Using a Multiple-input Residual Neural Network](https://mlsanity.com/paper/2412.10656/) (2024) ## Abstract (arXiv) 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-B'enard 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.