# An Analytical and AI-discovered Stable, Accurate, and Generalizable Subgrid-scale Closure for Geophysical Turbulence

arXiv: https://arxiv.org/abs/2509.20365 · HTML: https://mlsanity.com/paper/2509.20365/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 92 references that are in the map.
- First posted on arXiv 5 Sep 2025.

## Builds on (most cited first, 10 of 10)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Scientific multi-agent reinforcement learning for wall-models of turbulent flows](https://mlsanity.com/paper/2106.11144/) (2021)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Explaining the physics of transfer learning a data-driven subgrid-scale closure to a different turbulent flow](https://mlsanity.com/paper/2206.03198/) (2022)
- [Deep Reinforcement Learning for Turbulence Modeling in Large Eddy Simulations](https://mlsanity.com/paper/2206.11038/) (2022)
- [Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challenges](https://mlsanity.com/paper/2306.05014/) (2023)
- [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)
- [Coarse Graining with Neural Operators for Simulating Chaotic Systems](https://mlsanity.com/paper/2408.05177/) (2024)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)

## Built on by (most cited first, 1 of 1)

- [Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms](https://mlsanity.com/paper/2604.18414/) (2026)

## Abstract (arXiv)

By combining AI and fluid physics, we discover a closed-form closure for 2D turbulence from small direct numerical simulation (DNS) data. Large-eddy simulation (LES) with this closure is accurate and stable, reproducing DNS statistics including those of extremes. We also show that the new closure could be derived from a 4th-order truncated Taylor expansion. Prior analytical and AI-based work only found the 2nd-order expansion, which led to unstable LES. The additional terms emerge only when inter-scale energy transfer is considered alongside standard reconstruction criterion in the sparse-equation discovery.