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Online learning of eddy-viscosity and backscattering closures for geophysical turbulence using ensemble Kalman inversion

Yifei Guan*

Pedram Hassanzadeh†

Tapio Schneider and Oliver Dunbar

Daniel Zhengyu Huang

Jinlong Wu

Ignacio Lopez-Gomez

  • Department of Geophysical Sciences and Committee on Computational and Applied Mathematics, The University of Chicago, Chicago, Illinois, USA

  • Beijing International Center for Mathematical Research, Peking University, Beijing, China

  • *Contact author: guany@union.edu
  • †Contact author: pedramh@uchicago.edu
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Phys. Rev. Research 8, 023215 – Published 27 May, 2026

DOI: https://doi.org/10.1103/mnbm-3g56

Abstract

This article is part of a Physical Review Collection on the Physics of Changing Climate.

Different approaches to using data-driven methods for subgrid-scale closure modeling of geophysical turbulence have emerged recently. Most of these approaches are data hungry and lack interpretability and out-of-distribution generalizability. Here, we use a hybrid approach that combines turbulence theory, physics-based modeling, and data-driven methods to overcome these challenges. Specifically, we address the parametric uncertainty of well-known physics-based large-eddy simulation (LES) closures: the Smagorinsky (Smag) and Leith eddy-viscosity models (one free parameter) and the Jansen-Held (JH) backscattering model (two free parameters). For various cases of two-dimensional turbulence, optimal parameters are first learned online from data via ensemble Kalman inversion (EKI), such that for each case, the LES energy spectrum matches that of direct numerical simulation (DNS). We quantify the uncertainties on these parameters using a modern machine-learning-accelerated Bayesian workflow, “Calibrate, Emulate, Sample.” Only a small training dataset is needed (to calculate the DNS spectra); i.e., the approach is data-efficient. We find the optimized parameter(s) and their associated uncertainty for each closure to be constant across broad flow regimes that differ in dominant length scales, eddy/jet structures, and dynamics, suggesting that these closures are generalizable. Next, we show that the online learned constants agree with the predictions of a recent semianalytical derivation, providing further interpretability. In both a priori and a posteriori tests that include examining the extreme events, LES with optimized closures, especially with JH, outperforms the baselines (LES with standard Smag, dynamic Smag, or Leith). This work shows the promise of combining advances in theory, physics-based modeling (e.g., JH), and data-driven modeling (e.g., online learning with EKI) to develop data-efficient frameworks for accurate, interpretable, and generalizable closures for geophysical turbulence, with ultimate applications in weather and climate prediction.

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This article appears in the following collection:

Physics in a Changing Climate

Climate Science is a growing field that is attracting colleagues from many disciplines. This Collection, a joint effort by Physical Review E, Physical Review Research, and PRX Energy curated by Prof. Justin Burton from Emory University, aims at providing a survey of the diverse range of topics that scientists with a connection to Physics are exploring in this exciting area of research. This includes climate and related geosciences, sustainability, energy, and material science, and may include fundamental as well as applied research.

Every article in this Collection was subjected to a rigorous peer review process, upholding the high standards applied to all papers. The Physical Review E, Physical Review Research, and PRX Energy editorial teams managed the peer review and made all editorial decisions.

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Phys. Rev. Research 8, 023215– Published 27 May, 2026
  • Received 9 May 2025
  • Accepted 3 May 2026
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